NASA 

Reference 

Publication 

1152 

AVSCOM 

Technical 

Report 

84-C-1 5 

1985 

NatIonal Aeronauilcs 

and Space Admlntstratlon 

Scientific and Technical 

Information Branch 

Gearing 

John J. Coy 

Propulsion Directorate 

USAARTA-AVSCOM 

Lewis Research Center 

Cleueland, Ohio 

Dennis P. Townsend 

and Erwin V. Zaretsky 

Lewis Research Center 

Cleveland, Ohio 
Contents 

Page 

Summary .............................................................................................................. 1 

1 .O Introduction ..................................................................................................... 1 

1.1 Early History of Gearing ................................................................................. 1 

1.2 Beginning of Modern Gear Technology ............................................................... 2 

2.0 Types and Geometry.. ......................................................................................... 2 

2.1 Parallel-Shaft Gears.. ..................................................................................... 2 

2.2 Intersecting-Shaft Gears.. ................................................................................ 9 

2.3 Nonparallel-Nonintersecting-Shaft Gears ............................................................ 12 

3 .O Processing and Manufacture ................................................................................. 17 

3.1 Materials.. ................................................................................................... 17 

3.2 Metallurgical Processing Variables .................................................................... 23 

3.3 Manufacturing .............................................................................................. 2.5 

4.0 Stresses and Deflections.. ..................................................................................... 26 

4.1 Lewis Equation Approach for Bending Stress.. ..................................................... 26 

4.2 Allowable Bending Stress ................................................................................ 3 1 

4.3 Other Methods ............................................................................................. 3 1 

,5.0 Gear Life Predictions.. ........................................................................................ 34 

5.1 Theory of Gear Tooth Life .............................................................................. 35 

5.2 Gear Life.. ................................................................................................... 37 

5.3 Gear System Life.. ......................................................................................... 37 

5.4 Helical Gear Life ........................................................................................... 38 

5.5 Bevel and Hypoid Gear Life ............................................................................. 38 

6.0 Lubrication ...................................................................................................... 40 

6. I Lubricant Selection.. ...................................................................................... 40 

6.2 Elastohydrodynamic Film Thickness .................................................................. 41 

6.3 Boundary Lubrication .................................................................................... 44 

6.4 Lubricant Additive Selection.. .......................................................................... 44 

6.5 Jet Lubrication ............................................................................................. 45 

6.6 Gear Tooth Temperature ................................................................................. 47 

7.0 Power Loss Predictions ....................................................................................... 47 

7.1 Sliding Loss ................................................................................................. 48 

7.2 Rolling Loss ................................................................................................. 49 

7.3 Windage Loss.. ............................................................................................. 49 

7.4 Other Losses ................................................................................................ 50 

7.5 Optimizing Efficiency ..................................................................................... 50 

8.0 Optimal Design of Spur Gear Mesh ........................................................................ 5 1 

8.1 Minimum Size .............................................................................................. 51 

8.2 Specific Torque Capacity ................................................................................ 52 

9.0 Gear Transmission Concepts ................................................................................ 53 

9.1 Series Trains ................................................................................................ 53 

9.2 Multispeed Gears .......................................................................................... 54 

9.3 Epicyclic Gears ............................................................................................. 54 

9.4 Split-Torque Transmissions ............................................................................. 57 

9.5 Differential Gearing ....................................................................................... 57 

9.6 Closed-Loop Trains ....................................................................................... 58 

Appendix-Symbols ............................................................................................... 59 

References ............................................................................................................ 62 

Bibliography ......................................................................................................... 66 

... 111 
Summary 

Gearing technology in its modern form has a history of 

only 100 years. However, the earliest form of gearing can 

probably be traced back to fourth century B.C. Greece. 

This publication draws together current gear practice and 

integrates it with recent advances in the technology. The 

history of gearing is reviewed briefly in the “Introduction.” 

Subsequent sections describe types of gearing and their 

geometry, processing, and manufacture. Both conventional and more recent methods of determining gear 

stress and deflections are considered. The subjects of life 

prediction and lubrication are additions to the literature, 

not having been treated before. New and more complete 

methods of power loss prediction as well as an optimum 

design of spur gear meshes are described. Conventional 

and new types of power transmission system are presented. 

1.0 Introduction 

Gears are the means by which power is transferred 

from source to application. Gearing and geared transmissions drive the machines of modern industry. Gears 

move the wheels and propellers that transport us over the 

sea, on the land, and in the air. A sizable section of 

industry and commerce in today’s world depends on 

gearing for its economy, production, and livelihood. 

The art and science of gearing have their roots before 

the common era. Yet many engineers and researchers 

continue to delve into the areas where improvements are 

necessary, seeking to quantify, establish, and codify 

methods to make gears meet the ever-widening needs of 

advancing technology (ref. 1). 

1.1 Early History of Gearing 

The earliest written descriptions of gears are said to 

have been made by Aristotle in the fourth century B.C. 

(ref. 2). It has been pointed out (refs. 3 and 4) that the 

passage attributed by some to Aristotle, in “Mechanical 

Problems of Aristotle” (ca. 280 B.C.), was actually from 

the writings of his school. In the passage in question there 

was no mention of gear teeth on the parallel wheels, and 

they may just as well have been smooth wheels in 

frictional contact. Therefore the attribution of gearing to 

Aristotle is most likely incorrect. The real beginning of 

gearing was probably with Archimedes, who in about 250 

B.C. invented the endless screw turning a toothed wheel, 

which was used in engines of war. Archimedes also used 

gears to simulate astronomical ratios. The Archimedian 

spiral was continued in the hodometer and dioptra, which 

were early forms of the wagon mileage indicator 

(odometer) and the surveying instrument. These devices 

were probably based on “thought” experiments of 

Heron of Alexandria (ca. 60 A.D.), who wrote on 

theoretical mechanics and the basic elements of 

mechanisms. 

The oldest surviving relic containing gears is the 

Antikythera mechanism, named for the Greek island near 

which the mechanism was discovered in a sunken ship in 

1900. Professor Price of Yale University has written an 

authoritative account of this mechanism (ref. 3). The 

mechanism is not only the earliest relic of gearing, but is 

also an extremely complex arrangement of epicyclic 

differential gearing. The mechanism is identified as a 

calendrical Sun and Moon computing mechanism and is 

dated to about 87 B.C. 

The art of gearing was carried through the European 

Dark Ages, appearing in Islamic instruments such as the 

geared astrolabes that were used to calculate the positions 

of the celestial bodies. Perhaps the art was relearned 

by the clock- and instrument-making artisans of 14th 

century Europe, or perhaps some crystallizing ideas and 

mechanisms were imported from the East after the 

crusades of the 11th through the 13th centuries. 

It appears that an English abbot of St. Alban’s 

monastery, born Richard of Wallingford in 1330 A.D., 

reinvented the epicyclic gearing concept. He applied it to 

an astronomical clock that he began to build and that was 

completed after his death. 

A mechanical clock of a slightly later period was 

conceived by Giovanni de Dondi (1348-1364). Diagrams 

of this clock, which did not use differential gearing 

(ref. 3), appear in the sketchbooks of Leonardo da Vinci, 

who designed geared mechanisms himself (refs. 3 to 5). In 

1967, two of Leonardo da Vinci’s manuscripts, lost in the 

National Library in Madrid since 1830, were rediscovered 

(ref. 6). One of the manuscripts, written between 1493 

and 1497 and known as “Codex Madrid I” (ref. 7), 

contains 382 pages with some 1600 sketches. Included 

among this display of Leonardo’s artistic skill and 

engineering ability are his studies of gearing. Among 

these are tooth profile designs and gearing arrangements 

that were centuries ahead of their “invention.” 
1.2 Beginning of Modern Gear Technology 

In the period 1450 to 1750 the mathematics of geartooth profiles and theories of geared mechanisms became 

established. Albrecht Diirer is credited with discovering 

the epicycloidal shape (ca. 1525). Philip de la Hire is said 

to have worked out the analysis of epicycloids and 

recommended the involute curve for gear teeth (ca. 1694). 

Leonard Euler worked out the law of conjugate action 

(ca. 1754) (ref. 5). Gears designed according to this law 

have a steady speed ratio. 

Since the industrial revolution in the mid 19th century, 

the art of gearing has blossomed, and gear designs have 

steadily become based on more scientific principles. In 

1893, Wilfred Lewis published a formula for computing 

stress in gear teeth (ref. 8). This formula is in wide use 

today in gear design. In 1899, George B. Grant, the 

founder of five gear manufacturing companies, 

published “A Treatise on Gear Wheels” (ref. 9). New 

inventions led to new applications for gearing. For 

example, in the early part of this century (1910), parallelIn 1976, bevel gears produced to drive a compressor 

test stand ran successfully at 2984 kW (4000 hp) and 203 

m/s (40 000 ft/min) for 235 hr (ref. 11). From all 

indications these gears could be used in an industrial 

application. A reasonable maximum pitch-line velocity 

for commercial spiral-bevel gears with curved teeth is 

60 m/s (12 000 ft/min) (ref. 12). 

Gear system development methods have been advanced 

in which lightweight, high-speed, highly loaded gears are 

used in aircraft applications. The problems of strength 

and dynamic loads, as well as resonant frequencies for 

such gearing, are now treatable with techniques such as 

finite-element analysis, siren and impulse testing for 

mode shapes, and damping (ref. 13). The material 

presented herein will assist in the design, selection, 

application, and evaluation of gear drives. Sizing criteria, 

lubricating considerations, material selection, and 

methods to estimate service life and power loss are 

presented. 

shaft gears were introduced to reduce the speed of the 

newly developed reaction steam turbine enough for it to 2.0 Types and Geometry 

turn the driving screws of ocean-going vessels. This 

application achieved an overall increase in efficiency of 

25 percent in sea travel (ref. 2). 

The need for more accurate and quieter running gears 

became obvious with the advent of the automobile. 

Although the hypoid gear was within our manufacturing 

capabilities by 1916, it was not used practically until 

1926, when it was used in the Packard automobile. The 

hypoid gear made it possible to lower the drive shaft and 

gain more usable floor space. By 1937 almost all cars 

used hypoid-geared rear axles. Special lubricant antiwear additives that made it practical to use hypoid 

gearing were formulated in the 1920’s. In 1931, Earle 

Buckingham, chairman of an ASME research committee 

on gearing, published a milestone report on the dynamic 

loading of gear teeth (ref. 10). This led to a better 

understanding of why faster-running gears sometimes 

could not carry as much load as slower-running gears. 

References 14 and 15 outline the various gear types and 

include information on proper gear selection. These 

references classify single-mesh gears according to the 

arrangement of their shaft axes in a single-mesh gearset 

(table 1). These arrangements are parallel shafts; 

intersecting shafts; and nonparallel, nonintersecting 

shafts. 

2.1 Parallel-Shaft Gears 

Spur gears.-The external spur gear (figs. 1 and 2) is 

the most common type of gear. Because the teeth are 

straight and parallel to the shaft axis, it is also the 

simplest type of gear. The smaller of two gears in mesh is 

called the pinion. The larger is customarily designated as 

“the gear.” In most applications the pinion is the driving 

element and the gear is the driven element. Most spur 

gear tooth profiles are cut to conform to an involute 

curve in order to ensure conjugate action. Conjugate 

action is defined as a constant angular velocity ratio 

between two meshing gears. Conjugate action can be 

obtained with any tooth profile shape for the pinion, 

provided that the mating gear is made with a tooth shape 

that is conjugate to the pinion tooth shape. The conjugate tooth profiles are such that the common normal at 

the point of contact between the two teeth will always 

pass through a fixed point on the line of centers. The 

fixed point is called the pitch point. 

High-strength alloy steels for gearing were developed 

during the 1920’s and 1930’s. Nitriding and casehardening techniques to increase the surface strength of 

gearing were introduced in the 1930’s. Induction 

hardening was introduced in 1950. Extremely clean steels 

produced by vacuum melting processes introduced in 

1960 have proven effective in prolonging gear life. 

Since the early 1960’s there has been increased use of 

industrial gas turbines for electric power generation. 

Epicyclic gear systems in the range 746 to 10 445 kW 

(1000 to 14 000 hp) have been used successfully. Pitchline velocities are from 50 to 100 m/s (IO 000 to 20 000 

ft/min). These gearsets must work reliably for 10 000 to 

30 000 hr between overhauls (ref. 1). 

The line of action, or pressure line (fig. 2), is the line 

that is tangent to both base circles. All points of contact 

between the two teeth will lie along this line. The pressure 

angle cp is defined as the acute angle between the line of 

2 
TABLE l.-SIMPLE MESH GEARS 

Type Nominal gear Maximum Nominal efficiency Remarks 

ratio range pitch-line at rated power, 

velocity percent 

m/s ft/min 

Parallel shafts 

Spur 

Helical 

Double helical 

(herringbone) 

Conformal 

l-5 

I-10 

I-20 

l-12 

a100 a20 000 97-99.5 

“20 b4 000 

a200 =4oooo 15”-35” Helix angle 

b2o “4 000 

a200 a40 000 

b2o ‘=4 000 

-__--- I :---- 

Intersecting shafts 

Straight bevel l-8 

Spiral bevel I-8 

ZerolC bevel 1-8 

a.50 a10 000 

“5 “1 000 

a125 a25 000 

“10 “12 000 

a50 a10 000 

t-3 “1 000 

97-99.5 

97-99.5 

97-99.5 

35” Spiral angle 

most used 

--__-------__---- 

Nonintersecting, nonparallel shafts 

Worm 3-100 a50 a10 000 

b25 9 000 

Double-enveloping 3-100 a50 a10 000 

worm b20 b4 000 

Crossed helical l-100 “50 a10 000 

b2o b4 000 

Hypoid I-50 a50 a10 000 

b2o b4 ooo 

aAircrafl high precismn. 

kommerciat. 

‘Regi\lered Iradcmark. The Gleason Works, Rochester. N\ 

50-90 

50-98 

50-95 

50-98 

--___------_----- 

--_-------------- 

--___------__---- 

Higher ratios have 

low efficiency 

Figure 1 .-External spur gears. 

action and a line perpendicular to the line of centers. 

Although the pressure angle can vary for different 

applications, most spur gears are cut to operate at 

pressure angles of 20” or 25”. The circular pitch p of a 

spur gear is defined as the distance on the pitch circle 

from a point on a tooth to the corresponding point on an 

adjacent tooth: 

p=% 

where 

rp pitch circle radius 

N number of teeth 

(1) 
01 f 

A 

u1 

,,- Pitch circle Pinion 

,- Base circle i I\ 

circle file is not an involute J 

lb1 

(a) Geometry. 

(b) Interference, on flank of pinion during approach action and flank of 

gear during recess action. 

Figure 2.-External spur gear geometry and interference in the action of 

gear teeth. 

Similarly the base pitch pb is the distance on the base 

circle from a point on one tooth to the corresponding 

point on an adjacent tooth: 

hrb 

Pb=-- N (2) 

where ‘b is the base circle radius. Also, from geometry 

Pb =P cos Q (3) 

The diametral pitch P, defined as the number of gear 

teeth N divided by the diameter of the pitch circle D, 

determines the relative sizes of gear teeth: 

p+ (4) 

where 

Pp=7r (5) 

The larger the value of P, the greater the number of gear 

teeth and the smaller their size. “Module” is the 

reciprocal of diametral pitch in concept, but whereas 

diametral pitch is expressed as the number of teeth per 

inch, the module is generally expressed in millimeters per 

tooth. Metric gears (in which tooth size is expressed in 

modules) and American standard inch diametral pitch 

gears are not interchangable. 

The distance between the centers of two gears (fig. 2) is 

(-Jr Dl +D2 

2 (6) 

The space between the teeth must be larger than the 

mating tooth thickness in order to prevent jamming of 

the gears. The difference between tooth thickness and 

tooth space as measured along the pitch circle is called 

backlash. Backlash can be created by cutting the gear 

teeth slightly thinner than the space between teeth, or by 

setting the center distance slightly greater between the 

two gears. In the second case the operating pressure angle 

of the gear pair is increased accordingly. The backlash 

for a gear pair must be sufficient to permit free action 

under the most severe combination of manufacturing 

tolerances and operating temperature variations. 

Backlash should be small in positioning control systems, 

but it should be quite generous for single-direction power 

gearing (table 2). If the center distance between mating 
TABLE 2.-RECOMMENDED 

BACKLASH FOR 

ASSEMBLED GEARS 

m,= z 

pb 

(11) 

Diametral pitch, 

P 

m-l 

25.4 1 1.064-O. 10 

17 1.5 .046-.O?O 

12.7 2 .036-.050 

10 2.5 .028-,040 

8.5 3 .024-.036 

6.4 4 .018-.028 

5 5 .015-,023 

4.2 6 .013-.020 

3.6 7 .OlO-,018 

3.2-2.8 8-9 .OlO-.015 

2.5-2 10-13 .008-,013 

1.8-.8 14-32 .005-,010 

in-’ 

T 

t 

Backlash 

cm 

1 

in 

X025-0.040 

.018-.027 

.014-.020 

.Ol lb.016 

.009-,014 

.007-.Oll 

.006-,009 

.005-.008 

.004-,007 

.00&.006 

.003-.005 

.002-,004 

external gears is increased by AC, the resulting increase in 

backlash AB is 

AB=ACx2tancp (7) 

For the mesh of internal gears this equation gives the 

decrease in backlash for an increase in center distance. 

The distance between points A and B (fig. 2), or the 

length of contact along the line of action, is the distance 

between the intersections, with the line of action, of the 

gear addendum circle and the pinion addendum circle. 

This is the total length along the line of action for which 

there is tooth contact. This distance can be determined by 

the various radii and the pressure angle as follows: 

z= [ I$, -($,I cm cp) 2 l/2 1 

+ [ 2,2 - (Q&2 cm P> 2 I 1’2-Csin cp 03) 

The radii of curvature of the teeth when contact is at a 

distance s along the line of action from the pitch point 

can be represented for the pinion as 

p1 =rp,l sin cpfS 

and for the gear as 

p2 = rp,2 sin co *S 

Where s is positive for equation (9), it must be negative 

for equation (10) and vice versa. 

To determine how many teeth are in contact, the 

contact ratio of the gear mesh must be known. The 

contact ratio m, is defined as 

Gears are generally designed with contact ratios of 1.2 to 

1.6. A contact ratio of 1.6, for example, means that 40 

percent of the time one pair of teeth will be in contact and 

60 percent of the time two pairs of teeth will be in 

contact. A contact ratio of 1.2 means that 80 percent of 

the time one pair of teeth will be in contact and 20 percent 

of the time two pairs of teeth will be in contact. 

Gears with contact ratios greater than 2 are referred to 

as “high-contact-ratio gears.” For these gears there are 

never less than two pairs of teeth in contact. A contact 

ratio of 2.2 means that 80 percent of the time two pairs of 

teeth will be in contact and 20 percent of the time three 

pairs of teeth will be in contact. High-contact-ratio gears 

are generally used in select applications where long life is 

required. Figure 3 shows a high-contact-ratio gear and a 

conventional involute gear. Analyses should be 

performed when using high-contact-ratio gearing because 

higher bending stresses may occur in the tooth addendum 

region. Also, higher sliding in the tooth contact can 

contribute to distress of the tooth surfaces. In addition, 

higher dynamic loading may occur with high-contactratio gearing (ref. 16). 

Interference of the gear teeth is an important 

consideration. The portion of the spur gear below the 

base circle is sometimes cut as a straight radial line but 

never as an involute curve (fig. 2(b)). Hence, if contact 

should occur below the base circle, nonconjugate action 

(interference) will occur. The maximum addendum 

radius of the gear without interference is calculated from 

the following equation: 

f-i= d++sinQ ( > 

l/2 (12) 

where r; is the radial distance from the gear center to the 

point of tangency of the mating gear’s base circle with the 

line of action. Should interference be indicated, there are 

several methods to eliminate it. The center distance can 

be increased, which will also increase the pressure angle 

and change the contact ratio. Also, the addendum can be 

shortened, with a corresponding increase in the 

addendum of the mating member. The method used 

depends on the application and experience. 

Another type of spur gear, internal, has the teeth cut 

on the inside of the rim (fig. 4). The pitch relationships 

previously discussed for external gears apply also to 

internal gears (eqs. (1) to (5)). A secondary type of 

involute interference called tip fouling may also occur. 

The geometry should be carefully checked in the region 

labeled “F” in figure 4 to see if this occurs. 
(b) c -78-4330 

(a) High-contact-ratio gear with modified tooth profile. Contact ratio, 2.25. 

(b) Conventional involute gear. Contact ratio, 1.3. 

Figure 3.-Comparison of gears. 

Pitch point 7 

\ r C learance 

/ 

Line of action \, \ / 

(pressure line) -’ .*- 

Circular pitch7 

Base circle of 

internal gear 

Circular 

thickness 

internal diameter 

Working depth 

I 

Whole depthA 

\ “\L Tooth flank 

\ \ 

L Tooth face 

Figure 4.-Internal spur gear geometry. 

Helical gears.-In a helical gearset (fig. 5) cylindrical 

gears have teeth cut in the form of a hehx: one gear has a 

right-hand helix and the mating gear, a left-hand helix. 

Helical gears have a greater load-carrying capacity than 

equivalent-size spur gears. Although helical gears vibrate 

less than spur gears because the teeth overlap, a high 

thrust load is produced along the axis of rotation. This 

load results in high rolling-element-bearing loads, which 

between 

C 

may reduce the life and reliability of a transmission 

system. This end thrust increases with helix angle. 

To overcome the axial thrust load, double-helical 

gearsets are used (fig. 6). The thrust loads are 

counterbalanced so that no resultant axial load is 

transmitted to the bearings. Space is sometimes provided 

between the two sets of teeth to allow for runout of the 

tooth cutting tool, or the gear may be assembled in two 

6 
Figure 5.--Single-helical gear pair. 

Double-helical teeth 

Figure 6.-Double-helical, or herringbone, gear pair. 

halves. The space should be small because it reduces the 

active face width. Double-helical, or herringbone, gears 

are used for the transmission of high torques at high 

speeds in continuous service. 

Spur gears have only one diametral and one circular 

pitch; helical gears have two additional pitches. The 

normal circular pitch pn is the distance between 

corresponding points of adjacent teeth as measured in 

plane B-B (fig. 7), which is perpendicular to the helix. 

Basic rack 

Section A-A (transverse plane) 

Section B-B (normal plane) 

Figure 7.-Helical gear geometry. (Not to scale.). 

pitch pa is a similar distance measured in a plane parallel 

to the shaft axis: 

pa =pt cot * (14) 

The diametral pitch in the transverse plane is 

pn =pt cos \k (13) 

where 

pt transverse circular pitch 

\k helix angle 

The transverse circular pitch is measured in plane A-A 

(fig. 7), which is perpendicular to the shaft axis. The axial 

P*=$ 

The normal diametral pitch is 

pt P,’ - 

cos (p 

(15) 

(16) 

7 
To maintain contact across the entire tooth face, the 

minimum face width F must be greater than the axial 

pitch pa. However, to ensure a smooth transfer of load 

during tooth engagement, F should be at least 1.2 to 2 

times pa. The two pressure angles associated with the 

helical gear, the transverse pressure angle pt measured in 

plane A-A (fig. 7) and the normal pressure angle (Ok 

measured in plane B-B, can be related as follows: 

tan pn = tan pt cos \k (17) 

The three components of normal load Wj,, acting on a 

helical gear tooth can be written as follows: 

Tangential: 

w* = w, cos (on cos 3f 

Radial: 

(18) 

Wr= W~sin pn (19) 

Axial: 

W, = W, cos p,, sin f cm 

where W, is the tangential load acting at the pitch circle 

perpendicular to the axis of rotation. The direction of the 

axial thrust load for pairs of helical gears (fig. 8) depends 

on whether the driver has a right-hand or left-hand helix 

and the direction of rotation. These loads must be 

considered when selecting and sizing the rolling-element 

bearings to support the shafts and gears. 

The plane normal to the gear teeth, B-B (fig. 7), 

intersects the pitch cylinder. The gear tooth profile 

generated in this plane would be a spur gear, with the 

same properties as the actual helical gear. The number of 

teeth of the equivalent spur gear in the normal plane is 

known as either the virtual or equivalent number of teeth. 

The equivalent number of teeth is 

N N,= - co&If 

Thrust 

(21) 

Conformal (Wildhaber/Novikov) gears.-In 1923, 

E. Wildhaber filed a U.S. patent application (ref. 17) for 

helical gearing of a circular arc form. In 1926, he filed a 

patent application for a method of grinding this form of 

gearing (ref. 18). Until Wildhaber, helical gears were 

made only with screw involute surfaces. However, 

Wildhaber gearing failed to receive the support necessary 

to develop it into a working system. In 1956, M.L. 

Novikov in the U.S.S.R. was granted a patent for a 

similar form of gearing (ref. 19). Conformal gearing then 

advanced in the U.S.S.R, but the concept found only 

limited application. Conformal gearing was also the 

subject of research in Japan and China but did not find 

widespread industrial application. 

The first and probably only aerospace application of 

conformal gearing was by Westland Helicopter in Great 

Britain (refs. 20 and 21). Westland modified the 

geometry of the conformal gears as proposed by Novikov 

and Wildhaber and successfully applied the modified 

geometry to the transmission of the Lynx helicopter. The 

major advantage demonstrated by this transmission system was increased gear load capacity without gear tooth 

fracture. It is anticipated that the gear system would 

exhibit improved power-to-weight ratio or increased life 

and reliability. However, these assumptions have neither 

been analytically shown nor experimentally proven. 

To achieve the maximum contact area in a conformal 

gear pair (fig. 9), the radii of two mating surfaces rl and 

Q should be identical. However, this is not practical. 

Inaccuracies in fabrication will occur on the curvatures 

Pinion -j WI 

Line of centers 

Figure 9.-Conformal tooth geometry, 

Left-hand driven 

Thrust 

Left-hand driven Left-hand driven Left-hand driven 

rust 

Right-hand driven 

Figure S.-Direction of axial thrust load for helical gears on parallel shafts. 

8 
and in other dimensions such as the center distance. Such 

inaccuracies can cause stress concentrations on the edges 

and tips of the gear teeth. To prevent this edge loading, 

the radii are made slightly unequal and are struck from 

different centers. Although this slight mismatch 

decreases the stress concentrations, the contact surface 

area is also reduced and this amplifies the Hertzian 

contact stress. The resultant contact stresses depend on 

the helix angle. 

As a historical note, in early designs thermal expansion 

in gearcases and shafts was sufficient to affect the gear 

pair center distances, increasing the contact stresses and 

leading to early failure (ref. 22). 

As was discussed for involute gearing, a constant 

velocity ratio exists for spur gear tooth pairs at every 

angular position because the teeth “roll” over each 

other. However, with conformal gearing, there is only 

one angular position where a tooth is in contact with its 

opposite number. Immediately before and after that 

point there is no contact. To achieve a constant velocity 

ratio, the teeth must run in a helical form across the gear, 

and an overlap must be achieved from one tooth pair to 

the next as the contact area shifts across the gear, from 

one side to the other, during meshing (ref. 21). The sweep 

velocity of the contact area across the face width is often 

thought of as being a pure rolling action, as no physical 

translation of the metal occurs in this direction. 

However, there is a small sliding component acting along 

the tooth. If the pitch surfaces of the mating gears are 

considered, the length of the meshing helices would be 

the same on both the pinion and the gear. But the 

conformal system uses an all-addendum pinion and alladdendum wheel; so when the contact area traverses one 

axial pitch, it must sweep over a longer distance on the 

pinion and a shorter distance on the gear. Thus a small 

sliding component exists (fig. 10(a)). 

The magnitude of the sliding can vary with tooth 

design. However, since the sliding velocity depends on 

the displacement from the pitch surface, the percentage 

of sliding will vary across the contact area (fig. 10(b)). 

The main sliding velocity, which occurs at the tooth 

contact, acts up and down the tooth height. The 

magnitude of this sliding component can vary up to 

approximately 15 percent of the pitch helix sweep 

velocity. Thus the slide-to-roll ratio seems to indicate why 

slightly lower power losses than with involute gears can 

be attained by using conformal gears (ref. 21). 

2.2 Intersecting-Shaft Gears 

Several types of gear system can be used for power 

transfer between intersecting shafts. The most common 

of these are straight-bevel gears and spiral-bevel gears. In 

addition, there are special bevel gears that accomplish the 

same or similar results and are made with special 

geometrical characteristics for economy. Among these 

(b) 

(a) Sliding components in rolling direction. 

(b) Relative sliding velocities in contact area. 

Figure lO.-Tooth contact trace diagram. 

are Zerol’ gears, Coniflex’ gears, Formate gears, 

Revacycle’ gears, and face gears. For the most part the 

geometry of spiral-bevel gears is extremely complex and 

does not lend itself to simplified formulas or analyses. 

The gear tooth geometry is dictated by the machine tool 

used to generate the gear teeth and the machine tool 

settings. 

Straight-bevel gears.-Bevel gears can be arranged in 

various ways (fig. 11(a)). A cutaway drawing of a 

straight-bevel gear (fig. I 1 (b)) shows the terminology and 

important physical dimensions. Straight-bevel gears are 

used generally for relatively low-speed applications with 

pitch-line velocities to 5.08 m/s (1000 ft/min) and where 

vibration and noise are not important criteria. However, 

with carefully machined and ground straight-bevel gears 

it may be possible to achieve speeds to 76.2 m/s (15 000 

ft/min). 

Bevel gears are mounted on intersecting shafts at any 

desired angle, although 90” is most common. They are 

designed and manufactured in pairs and as a result are 

not always interchangeable. One gear is mounted on the 

lRegistered trademark. The Gleason Works, Rochester, N.Y. 
Usual form Mike gear 

Mitre gear Mitre gear 

Crown gear Internal bevel gear 

Ial 

Aoex of pitch cone--/’ 

Fat 

Root cone -__ _ 

5 Addendum 

Dedendum / 

(b. 

\ Outside pyh diameter.rl 

_I 

i 

Pitch 

/ 

radius of 

equivalent spur gear 

(a) Gear arrangements. 

(b) Gear terminology. 

(c) Gear pair. 

Figure 11 .-Straight-bevel gear. 

10 
cantilevered, or outboard, end of the shaft. Because of 

the outboard mounting the deflection of the shaft where 

the gear is attached can be rather large. This would result 

in the teeth at the small end moving out of mesh. The 

load would thus be unequally distributed, with the larger 

ends of the teeth taking most of the load. To reduce this 

effect, the tooth face width is usually made no greater 

than one-third of the cone distance. 

Bevel gears are usually classified according to their 

pitch angle. A bevel gear having a pitch angle of 90” and 

a plane for its pitch surface is known as a crown gear. 

When the pitch angle of a bevel gear exceeds 90”, it is 

called an internal bevel gear. Internal bevel gears cannot 

have a pitch angle very much greater than 90” because of 

problems incurred in manufacturing such gears. These 

manufacturing difficulties are the main reason why 

internal bevel gears are rarely used. Bevel gears with pitch 

angles less than 90” are the type most commonly used. 

When two meshing bevel gears have a shaft angle of 90” 

and have the same number of teeth, they are called mitre 

gears. In other words, mitre gears have a speed ratio of 1. 

Each of the two gears has a 45” pitch angle. 

The relationship (Tregold’s approximation) between 

the actual number of teeth for the bevel gear, the pitch 

angle, and the virtual or equivalent number of teeth is 

given by 

N N,= - 

cos r (22) 

where r is the pitch angle. The back cone distance 

becomes the pitch radius for the equivalent spur gear. 

Face gears have teeth cut on the flat face of the blank. 

The face gear meshes at right angles with a spur or helical 

pinion. When the shafts intersect, the face gear is known 

as an “on center” face gear. These gears may also be 

offset to provide a right-angle nonintersecting shaft 

drive. 

Coniffex gears are straight-bevel gears whose teeth are 

crowned in the lengthwise direction to accommodate 

small shaft misalignments. 

Formate gears have the gear member of the pair with 

nongenerated teeth, usually with straight tooth profiles. 

The pinion member of the pair has generated teeth that 

are conjugate to the rotating gear. 

Revacycle gears are straight-bevel gears generated by a 

special process, with a special tooth form. 

Spiral-bevel gears.-The teeth of spiral-bevel gears 

(fig. 12(a)) are curved and oblique. These gears are 

suitable for pitch-line velocities to 61 m/s (12 000 

ft/min). Ground teeth extend this limit to 125 m/s 

(25 000 ft/min). Spiral-bevel gears differ according to the 

methods of generating the gear tooth surfaces. Among 

these methods are the Gleason method, the Klingelnberg 

system, and the Oerlikon system. These companies have 

Ial 

h. ,-Circular pitch 

(a) Gear pair. 

(b) Cutting spiral gear teeth on basic crown rack. 

Figure 12.-Spiral-bevel gear. 

developed specified and detailed directions for the design 

of spiral-bevel gears that are related to the respective 

method of manufacture. However, some general 

considerations, such as the concept of pitch cones, 

generating gears, and conditions of force transmission, 

are common for the generation of all types of spiral-bevel 

gear (fig. 12(b)). The actual tooth gear geometry is 

dictated by the machine tool settings supplied by the 

respective manufacturers. 

The standard pressure angle (p for spiral-bevel gears is 

20” although 14.5” and 16” angles are used. The usual 

spiral angle $ is 35’. Because spiral-bevel gears are much 

stronger than similar-sized straight-bevel or Zerol’ gears 

(fig. 13), they can be used for large-speed-reduction-ratio 

applications at a reduced overall installation size. The 

hand of the spiral should be selected so as to cause the 

1 Registered trademark, The Gleason Works, Rochester, N.Y. 

11 
Figure 13.-Zero1 gear. 

gears to separate from each other during rotation and not 

to force them together, which might cause jamming. 

Thrust loads produced in operation with spiral-bevel 

gears are greater than those produced with straight-bevel 

gears. An analogy for the relationship that exists between 

straight- and spiral-bevel gears is that existing between 

spur and helical gears. Localized tooth contact is also 

achieved. This means that some mounting and load 

deflections can occur without a resultant load 

concentration at the ends of the teeth. 

The total force W, acting normal to the pinion tooth 

and assumed to be concentrated at the average radius of 

the pitch cone can be divided into three perpendicular 

components. These are the transmitted, or tangential, 

load Wt; the axial, or thrust, component W,; and the 

separating, or radial, component W,. The force W, can, 

of course, be computed from 

w*= z: 

rav 

(23) 

where 

T input torque 

rav radius of pitch cone measured at midpoint of tooth 

The forces W, and W, depend on the hand of the spiral 

and the direction of rotation. Thus there are four possible 

cases to consider. For a right-hand pinion spiral with 

clockwise pinion rotation or for a left-hand spiral with 

counterclockwise rotation, the equations are 

w,= c-$ (tan pn cos P + sin # sin P) (25) 

where 

$ spiral angle 

P pinion pitch angle 

pn normal tooth pressure angle 

The other two cases are a left-hand spiral with clockwise 

rotation and a right-hand spiral with counterclockwise 

rotation. For these two cases the equations are 

w,= c$q (tan pn sin P + sin $ cos r) 

w,= -J$ (tan pn cos r - sin $ sin r’) (27) 

and the rotation is observed from the input end of the 

pinion shaft. 

Equations (24) to (27) give the forces exerted by the 

gear on the pinion. A positive sign for either W, or W, 

indicates that the load is directed away from the cone 

center. The forces exerted by the pinion on the gear are 

equal and opposite. Of course, the opposite of an axial 

pinion load is a radial gear load, and the opposite of a 

radial pinion load is an axial gear load. 

Zero1 bevel gears are used for much the same 

applications as straight-bevel gears. The suggested 

minimum number of teeth is 14, one or more teeth should 

always be in contact, and the basic pressure angle is 20”, 

although angles of 22.5” or 25” are sometimes used to 

eliminate undercutting. 

2.3 Nonparallel-Nonintersecting-Shaft Gears 

Nonparallel, nonintersecting shafts lie in parallel 

planes and can be skewed at any angle between zero and 

90”. Among the gear systems used to transfer power 

between nonparallel, nonintersecting shafts are hypoid 

gears, crossed-helical gears, worm gears, Cone-Drive2 

worm gears, face gears, Spiroid gears, Planoid3 gears, 

Helicon3 gears, and Beveloid4 gears. 

Hypoid gears.-Hypoid gears (fig. 14) are very similar 

to spiral-bevel gears. The main difference is that their 

pitch surfaces are hyperboloids rather than cones. As a 

result their pitch axes do not intersect: the pinion axis 

being above or below the gear axis. In general, hypoid 

gears are most desirable for those applications involving 

2Registered trademark, Michigan Tool Co., Detroit, Michigan. 

w,= c$q (tan (Do sin r - sin $ cos r) (24) 3Registered trademark, Illinois Tool Works, Chicago, Illinois. 

4Registered trademark, Vinco Corporation, Detroit, Michigan. 

12 
Figure 14.-Hypoid gear. 

large speed-reduction ratios, those involving nonintersecting shafts, and those requiring great smoothness and 

quietness of operation. Hypoid gears are almost 

universally used for automotive applications, allowing 

the drive shaft to be located underneath the passenger 

compartment floor. They operate more smoothly and 

quietly than spiral-bevel gears and are stronger for a 

given ratio. Because the two supporting shafts do not 

intersect, bearings can be mounted on both sides of the 

gear to provide extra rigidity. The pressure angles usually 

range between 19” and 22.5”. The suggested minimum 

number of teeth is eight for speed ratios greater than 6 

and six for smaller ratios. High-reduction hypoids permit 

ratios between 10 and 120 and even as high as 360 in fine 

pitches. 

Crossed-helical gears.-Crossed-helical gears (fig. 15) 

are also known as spiral gears. They are in fact ordinary 

helical gears used in nonparallel-shaft applications. They 

transmit relatively small amounts of power because of 

their sliding action and limited tooth contact area. 

However, they permit a wide range of speed ratios 

without change of center distance or gear size. They can 

be used at angles other than 90”. 

For two helical gears to operate as crossed-helical gears 

they must have the same normal diametral pitch P,, and 

normal pressure angle pn. The gears do not need to have 

the same helix angle or be of opposite hand. In most 

crossed-helical gear applications the gears have the same 

hand. The relationship (fig. 16) between the helix angles 

of the gears and the angle between the shafts on which the 

gears are mounted is given by 

C=k,+\k2 (28) 

for gears having the same hand and by 

Figure 15 .-Crossed-helical gear. 

c=\kyP2 (29) 

for gears having opposite hands, where C is the shaft 

angle. For crossed-helical gears where the shaft angle is 

90” the helical gears must be of the same hand. 

The center distance between the gears is given by 

1 

c=2pn (- 

Nl I N2 

cos *‘1 cos *‘2 > 

(30) 

The sliding velocity vS acts at the pitch point tangentially 

along the tooth surface (fig. 16), where by the law of 

cosines 

( l/2 

us= iJ:+u~-2u~v2cos c 

> 

where 

v1= ~l’p,l 

~2 = w2rp,2 

For a shaft angle of 90” the sliding velocity is given by 

Ul v2 

us= - = - cos f2 cos \k, (32) 

The direction of the thrust load for crossed-helical 

gears can be determined from figure 17. The forces on the 

crossed-helical gears (fig. 16) are as follows: 

13 
Bottom 

Section A-A (normal plane 

Figure l&-Meshing crossed-helical gears showing relationship between helix angles, shaft angle, and load and velocity vectors. 

45’ delicals on right-angle shafts 

Thru Drive& Thruse Drive 

(a) (bl 

Driv 

hrust 

Driver 

- 

(a) Both gear and pinion left hand. 

(b) Both gear and pinion right hand. 

Figure 17.-Direction of thrust load for two meshing crossed-helical gears mounted on shafts oriented 90” to each other. 

14 
Transmitted force: 

w, = w, cos ‘Pn cos \k (33) 

Axial thrust load: 

W,= W, cos pn sin \k (34) 

Radial or separating force: 

W,= W, sin (Ok (35) 

where (Ok is the normal pressure angle. 

The pitch circle diameter D is obtained from the 

equation 

where 

N number of teeth 

P, normal diametral pitch 

‘$ helix angle 

Since the pitch diameters are not directly related to the 

number of teeth, they cannot be used to obtain the 

angular velocity ratio. This ratio must be obtained from 

the ratio of the number of teeth. 

Worm gears.-The worm gearset (fig. 18) comprises a 

worm, which resembles a screw, and a worm wheel, 

which is a helical gear. The shafts on which the worm and 

wheel are mounted are usually 90” apart. Because of the 

screw action of worm gear drives they are quiet and 

vibration free and produce a smooth output. On a given 

center distance much higher ratios can be obtained 

through a worm gearset than with other conventional 

types of gearing. The contact (Hertz) stresses are lower 

with worm gears than with crossed-helical gears. 

Although there is a large component of sliding motion in 

a worm gearset (as in crossed-helical gears), a worm 

gearset will have a higher load capacity than a crossedhelical gearset. 

Worm gearsets may be either single or double 

enveloping. In a single-enveloping set (fig. 19) the worm 

wheel has its width cut into a concave surface, thus 

partially enclosing the worm when in mesh. The doubleenveloping set, in addition to having the helical gear 

width cut concavely, has the worm length cut concavely. 

The result is that both the worm and the gear partially 

enclose each other. A double-enveloping set will have 

more teeth in contact and will have area rather than line 

contact, thus permitting greater load transmission. 

All worm gearsets must be carefully mounted to ensure 

proper operation. The double-enveloping, or cone, type 

is much more difficult to mount than the singleenveloping type. 

The relationships to define the worm gearset (figs. 18 

and 19) are as follows: the lead angle of the worm is 

LW Xw= arctan - 

TDW 

(37) 

This is the angle between a tangent to the pitch helix and 

the plane of rotation. If the lead angle is less than 5”, a 

rotation worm gearset cannot be driven backwards or, in 

other words, is self-locking. The pitch diameter is Dw 

The lead is the number of teeth or threads multiplied by 

the axial pitch of the worm or 

Lw=NwPa, w (38) 

The lead and helix angles of the worm and worm wheel 

are complementary 

Xw+Qw=90” (39) 

where qw is the helix angle of the worm. Also 

and for 90” shaft angles 

xw=*G (41) 

The center distance is given by 

Dw+DG c= 2 (42) 

where DG is the pitch diameter of the gear wheel. 

The American Gear Manufacturers Association 

(AGMA) recommends that the following equation be 

used to check the magnitude of Dw: 

D (43) 
I 1 rOutside diameter 

Addendum I+- Root ,~ 4 Whole depth 

alamerer 

4 I 

t t 

Addendum 

- Normal -v pressure 

\ fl angle, ‘Pn 

-II- Axial pressure 

Dedendum 1 

Axial 

Pitch 

diameter 

(a) Worm and wheel. 

(b) Worm. 

Figure 18.-Worm gearset nomenclature. 
r Worm 

Worm 

Figure 19.-Worm gearset showing load vectors. 

where PG is the circular pitch of the worm wheel. The 

gear ratio is given by 

If friction is neglected, the forces acting on the tooth 

surface (fig. 18) are as follows: 

(1) Transmitted force, which is tangential to the worm 

and axial to the worm wheel 

W,= WNcos qn sin hw (45) 

(2) Axial thrust load on the worm and tangential or 

transmitted force on the worm wheel 

w,= w,cos pn cos xw 

(3) Radial or separating force 

(46) 

W,= WNsin (Pi (47) 

The directions of the thrust loads for worm gearsets are 

given in figure 20. 

3 .O Processing and Manufacture 

3.1 Materials 

A wide variety of gear materials are available today for 

the gear designer. The designer can choose from wood, 

plastics, aluminum, magnesium, titanium, bronze, brass, 

gray cast iron, nodular and malleable iron, and a whole 

variety of low-, medium-, and high-alloy steels. In 

addition, there are many ways of modifying or processing 

the materials to improve their properties or to reduce the 

cost of manufacture. These include reinforcements for 

plastics, coating and processing for aluminum and 

titanium, and hardening and forging for many of the 

iron-based (or ferrous) gear materials. 

When selecting a gear material for an application, the 

gear designer will first determine the requirements for the 

gears being considered. The design requirements for a 

gear in a given application will depend on such conditions 

as accuracy, load, speed, material, and noise limitations. 

The more stringent these requirements are, the more 

costly the gear will become. For instance, a gear requiring 

high accuracy because of speed or noise limitations may 

require several processing operations in its manufacture. 

Each operation will add to the cost. Machined gears, 

17 
Thr 

Driver Driver 

Driver 

Th 

(al 

Driver 

Driver 

T 

Driver I 

Left-hand 

drivers 

(a) Right-hand drivers. 

(b) Left-hand drivers. 

Figure 20.-Direction of thrust load for worm gearset. 

which are the most accurate, can be made from materials 

with good strength characteristics. However, these gears 

are very expensive. The cost is further increased if 

hardening and grinding are required as in applications 

where noise limitation is a critical requirement. Machined 

gears should be a last choice for a high-production gear 

application because of cost. 

Some of the considerations in the choice of a material 

include allowable bending and Hertz stress, wear 

resistance, impact strength, water and corrosion 

resistance, manufacturing cost, size, weight, reliability, 

and lubrication requirements. Steel, under proper heat 

treatment, meets most of these qualifications. Where 

wear is relatively severe, as with worm gearing, a highquality chill-cast nickel bronze may be used for rim 

material. The smaller worm gears may be entirely of 

nickel bronze. 

18 

In aircraft applications, such as helicopter, V/STOL, 

and turboprops, the dominant factors are reliability and 

weight. Cost is of secondary importance. Off-the-road 

vehicles, tanks, and some actuators may be required to 

operate at extremely high loads with a corresponding 

reduction in life. These loads can produce bending 

stresses in excess of 1.03 GPa (150 000 psi) and Hertz 

stresses in excess of 2.76 GPa (400 000 psi) for portions 

of the duty cycle. This may be acceptable because of the 

relatively short life of the vehicle and a deemphasis on 

reliability. (In contrast, aircraft gearing typically operates 

at maximum bending stresses of 0.45 GPa (65 000 psi) 

and maximum Hertz stresses of 1.24 GPa (180 000 psi)). 

Plastics.-There has always been a need for a lightweight, low-cost gear material for light-duty applications. 

In the past, gears were made from wood or phenolicresin-impregnated cloth. However, in recent years, with 
the development of many new polymers, many gears are 

made of various “plastic” materials (table 3). The most 

common molded plastic gears are the acetate and nylon 

resins. These materials are limited in strength, 

temperature resistance, and accuracy. The nylon and 

acetate resins have a room-temperature yield strength of 

approximately 69 MPa (10 000 psi). This is reduced to 

approximately 28 MPa (4000 psi) at their upper 

temperature limit of 121 “C (250 “F). Nylon resin is 

subject to considerable moisture absorption, which 

reduces its strength and causes considerable expansion. 

Larger gears are made with a steel hub that has a plastic 

tire for better dimensional control. Plastic gears can 

operate for long periods in adverse environments, such as 

dirt, water, and corrosive fluids, where other materials 

would tend to wear excessively. They can also operate 

without lubrication or can be lubricated by the processed 

material as in the food industry. The cost of plastic gears 

can be as low as a few cents per gear for a simple gear on 

a high-volume production basis. This is probably the 

most economical gear available. Often a plastic gear is 

run in combination with a metal gear to give better 

dimensional control, low wear, and quiet operation. 

Polyimide is a more expensive plastic material than 

nylon or acetate resin, but it has an operating temperature limit of approximately 3 16 “C (600 “F). This makes 

the polyimides suitable for many adverse applications that 

might otherwise require metal gears. Polyimides can be 

used effectively in combination with a metal gear without 

lubrication because of polyimide’s good sliding properties 

(refs. 23 to 25). However, in addition to the greater material cost, polymide gears are more expensive than other 

plastic gears because they are more difficult to mold. 

Nonferrous metals.-Several grades of wrought and 

cast aluminum alloys are used for gearing. The wrought 

alloy 7075-T6 is stronger than 2024-T4 but is also more 

expensive. Aluminum does not have good sliding and 

wear properties. However, it can be anodized with a thin, 

hard surface layer that will give it better operating 

characteristics. Anodizing gives aluminum good corrosion protection in saltwater applications, but the coating 

is thin and brittle and may crack under excessive load. 

Magnesium is not considered a good gear material 

because of its low elastic modulus and other poor 

mechanical properties. 

Titanium has excellent mechanical properties, 

approaching those of some heat-treated steels, with a 

density nearly half that of steel. However, because of its 

poor sliding properties, which produce high friction and 

wear, it is not generally used as a gear material. Several 

attempts have been made to develop a wear-resistant 

coating, such as chromium plating, iron coating, or 

nitriding for titanium (refs. 26 to 29). Titanium would be 

an excellent gear material if a satisfactory coating or alloy 

could be developed to provide improved sliding and wear 

properties. 

Several alloys of zinc, aluminum, magnesium, brass, 

and bronze are used as die-cast materials. Many prior diecast applications now use less-expensive plastic gears. 

The die-cast materials have higher strength properties, 

are not affected by water, and will operate at higher 

temperatures than the plastics. As a result they still have a 

place in moderate-cost, high-volume applications. Diecast gears are made from copper or lower cost zinc or 

aluminum alloys. The main advantage of die casting is 

that it completely eliminates or drastically reduces the 

requirement for machining. The high fixed cost of the 

dies makes low production runs uneconomical. Some of 

the die-cast alloys used for gearing are listed in references 

30 and 31. 

Copper alloys.-Several copper alloys are used in gearing. Most are the bronze alloys containing varying 

amounts of tin, zinc, manganese, aluminum, 

phosphorus, silicon, lead, nickel, and iron. The brass 

alloys contain primarily copper and zinc with small 

amounts of aluminum, manganese, tin, and iron. The 

copper alloys are most often used in combination with an 

iron or steel gear to give good wear and load capacity 

especially in worm gear applications, where there is a 

high sliding component. In these cases the steel worm 

drives a bronze gear. Bronze gears are also used where 

corrosion and water are a problem. Several copper alloys 

are listed in table 4. The bronze alloys are either 

aluminum bronze, manganese bronze, silicon bronze, or 

TABLE 3.-PROPERTIES OF PLASTIC GEAR MATERIALS 

Property ASTM Acetate Nylon Polyimide 

Yield strength, MPa (psi) D 638 69 (10 000) 81 (11 800) 72 (IO 500) 

Shear strength, MPa (psi) D 732 66 (95 IO) 66 (9600) 81 (11 900) 

Impact strength (Izod), N-m (ft-lb) D 256 1.9 (1.4) 1.2 (0.9) 1.2 (0.9) 

Elongation at yield, percent D 638 15 5 6.5 

Modulus of elasticity, GPa (psi) D 790 2.8 (410 000) 2.8 (410 000) 3.2 (460 000) 

Coefficient of linear thermal D 696 8.8 x 10-s 8.1 x 1O-5 5.04x 10-s 

expansion, cm/cm “C (in/in “F) (4.5 x 10-X) (4.5 x 10-q (2.8 x 10m5) 

Water absorption (24 hr), percent D 570 0.25 1.5 0.32 

Specific gravity D 792 1.425 1.14 1.43 

Temperature limit, “C (“F) 121 (250) 121 (250) 315 (600) 

19 
TABLE 4.-PROPERTIES OF COPPER ALLOY 

GEAR MATERIALS 

Material T 

Yellow brass 104 15 x 106 345 50x 103 

Naval brass 104 15 310 45 

Phosphorous bronze 104 15 276 40 

Aluminum bronze 131 19 345 50 

Manganese bronze 110 16 310 45 

Silicon bronze 104 15 207 30 

Nickel-tin bronze 104 15 173 25 

Modulus of 

elasticity T 

GPa psi MPa 

Yield Ultimate 

strength strength 

psi 

phosphorous bronze. These bronze alloys have yield 

strengths of 138 to 414 MPa (20 000 to 60 000 psi) and all 

have good machinability. 

Cast iron.-Cast iron is used extensively in gearing because of its low cost, good machinability, and moderate 

mechanical properties. Many gear applications can use 

gears made from cast iron because of its good sliding and 

wear properties, which are in part a result of its free 

graphite and porosity. There are three basic cast irons, 

which are distinguished by the structure of graphite in the 

ferrite matrix: (1) gray cast iron, where the graphite is in 

flake form; (2) malleable cast iron, where the graphite 

consists of uniformly dispersed, fine, free-carbon 

particles or modules; and (3) ductile iron, where the 

graphite is in the form of tiny balls or spherules. The 

malleable iron and ductile iron have more shock and 

impact resistance. The cast irons can be heat treated to 

give improved mechanical properties. The bending 

strength of cast iron ranges from 34 to 172 MPa (5000 to 

25 000 psi) (ref. 32), and the surface fatigue strength 

ranges from 345 to 793 MPa (50 000 to 115 000 psi) 

(ref. 33). In many worm gear drives the bronze gear can 

be replaced with a cast iron gear at a lower cost because 

of the sliding properties of the cast iron. 

Sintered powder metals.-Sintering of powder metals 

is one of the more common methods used in highvolume, low-cost production of medium-strength gears 

with fair dimensional tolerance (ref. 34). In this method a 

fine metal powder of iron or other material is placed in a 

high-pressure die and formed into the desired shape and 

density under very high pressure. The unsintered (green) 

part has no strength as it comes from the press. It is then 

sintered in a furnace under a controlled atmosphere to 

fuse the powder together for greater strength and 

toughness. Usually an additive (such as copper in iron) is 

used in the powder for added strength. The sintering 

temperature is then set at the melting temperature of the 

copper to fuse the iron powder together for a stronger 

bond than would be obtained with the iron powder alone. 

The parts must be properly sintered to give the desired 

strength. 

Several materials with a wide range of properties are 

available for sintered powder-metal gears. Table 5 lists 

properties of some of the more commonly used gear 

TABLE 5.-PROPERTIES OF SINTERED POWDER-METAL ALLOY 

GEAR MATERIALS 

Composition 

1 to 5Cu-0.6C-94Fe 

7Cu-93Fe 

15Cu-0.6C-84Fe 

O.l5C-98Fe 

0.5C-96Fe 

2.5Mo-0.3C-1.7N-95Fe 

4Ni-lCu-0.25C-94Fe 

5Sn-95Cu 

lOSn-87Cu-0.4P 

1.5Be-0.25Co-98Cu 

Ultimate 

tensile 

strength 

- 

GPa 

- 

414 

221 

587 

359 

345 

897 

828 

138 

207 

552 

- 

psi 

60x 10’ B-60 Good impact strength 

32 B-35 Good lubricant impregnation 

85 B-80 Good fatigue strength 

52 A-60 Good impact strength 

50 B-75 Good impact strength 

130 c-35 High strength, good wear 

120 c-40 Carburized and hardened 

20 H-52 Bronze alloy 

30 H-75 Phosphorous bronze alloy 

80 B-85 Beryllium alloy 

20 
materials; other materials are available. The cost for 

volume production of sintered powder-metal gears is an 

order of magnitude lower than that for machined gears. 

A process that has been more recently developed is the 

hot-forming powder-metals process (refs. 35 and 36). In 

this process a powder-metal preform is made and 

sintered. The sintered powder-metal preformed part is 

heated to forging temperature and finished forged. The 

hot-formed parts have strengths and mechanical properties approaching the ultimate mechanical properties of 

the wrought materials. A wide choice of materials is 

available for the hot-forming powder-metals process. 

Since this is a fairly new process, there will undoubtedly 

be improvements in the materials made from this process 

and reductions in the cost. Several promising hightemperature, cobalt-base alloy materials are being 

developed. 

Because additional processes are involved, hot-formed 

powder-metal parts are more expensive than those 

formed by the sintered powder-metal process. However, 

either process is more economical than machining or 

conventional forging while producing the desired 

mechanical properties. This makes the hot-forming 

powder-metals process attractive for high-production 

parts where high strength is needed, such as in the 

automotive industry. The accuracy of the powder-metal 

and hot-forming processes is generally in the AGMA 

class 8 range. Although accuracy can be improved with 

special precautions if die wear is limited, this tends to 

increase the cost. Figure 21 shows the relative cost of 

some of the materials or processes for high-volume, lowcost gearing. 

Hardened steels. -A large variety of iron or steel alloys 

are used for gearing (table 6). The choice of which 

material to use is based on a combination of operating 

conditions such as load, speed, lubrication system, and 

temperature plus the cost of producing the gears. When 

operating conditions are moderate, such as medium loads 

with ambient temperatures, a low-alloy steel can be used 

without the added cost of heat treatment and additional 

processing. The low-alloy material in the non-heattreated condition can be used for bending stresses in the 

138-MPa (20 000-psi) range and surface durability Hertz 

stresses of approximately 586 MPa (85 000 psi). As the 

operating conditions become more severe, it becomes 

necessary to harden the gear teeth for improved strength 

and to case harden the gear tooth surface by case 

carburizing or case nitriding for longer pitting fatigue 

life, better scoring resistance, and better wear resistance. 

Several medium-alloy steeIs can be hardened to give good 

load-carrying capacity with bending stresses of 345 to 414 

MPa (50 000 to 60 000 psi) and contact stresses of 1.1 to 

1.24 GPa (160 000 to 180 000 psi). The higher alloy steels 

are much stronger and must be used in heavy-duty 

applications. AISI 9310, AISI 8620, Nitralloy N, and 

Super Nitralloy are good materials for these applications 

10-2 J/l-l 

Molded Stamp 

/ 

/ 

plastic metal melal forged 

powder 

metal 

Figure 21 .-Relative costs of gear materials. 

and can operate with bending stresses of 483 MPa (70 000 

psi) and maximum contact (Hertz) stresses of 1.38 GPa 

(200 000 psi). These high-alloy steels should be case 

carburized (AISI 8620 and 9310) or case nitrided 

(Nitralloy) for a hard, wear-resistant surface. 

Gears that are case carburized will usually require 

grinding after the hardening operation because of 

distortion during heat treatment. The nitrided materials 

offer the advantage of much less distortion during 

nitriding and therefore can be used in the as-nitrided 

condition without additional finishing. This is very 

helpful for large gears with small cross sections, where 

distortion can be a problem. Since case depth for 

nitriding is limited to approximately 0.051 cm (0.020 in), 

case crushing can occur if the load is too high. 

Gear surface fatigue strength and bending strength can 

be improved by shot peening (refs. 37 and 38). The 

lo-percent surface fatigue life of the shot-peened gears 

was 1.6 times that of the standard ground gears (ref. 37). 

The low- and medium-alloy steels have a limiting 

operating temperature above which they begin to lose 

their hardness and strength. Above this temperature, 

usually around 149 “C (300 “F), the materials are 

tempered and early bending failures, surface pitting 

failures, or scoring will occur. To avoid this condition, a 

material is needed that has a higher tempering temper- 
TABLE 6.-PROPERTIES OF STEEL ALLOY GEAR MATERIALS 

Material T 

Cast iron 310 45 x 103 ---- 

Ductile iron 552 80 414 

AISI 1020 552 80 483 

AISI 1040 690 100 414 

AISI 1066 828 120 621 

AISl 4146 667 140 883 

AISI 4340 932 135 828 

AISI 8620 1173 170 966 

AlSI 8645 1450 210 1242 

AISI 9310 1277 185 1104 

AISI 440C 759 110 499 

AlSl 416 1104 160 966 

AISI 304 759 110 518 

Nitralloy 135M 932 135 690 

Super Nitralloy 1450 210 1311 

CBS 600 1532 222 1242 

CBS IOOOM 1463 212 1201 

Vasco X-2 1711 248 1553 

EX-53 1180 171 973 

EX-14 1166 169 794 

Tensile 

strength T Yield strength 

MPa psi MPa psi 

60x lo3 

70 

60 

90 

128 

120 

140 

180 

160 

65 

140 

75 

100 

190 

180 

174 

125 

141 

115 

ature and that maintains its hardness at high temperatures. 

The generally accepted minimum hardness required at 

operating temperature is Rockwell C-58. In recent years 

several materials have been developed that maintain a 

Rockwell C-58 hardness at temperatures from 232 to 

315 “C (450 to 600 “F) (ref. 39). Several materials have 

shown promise of improved life at normal operating 

temperature. The hot-hardness data indicate that they 

will also provide good fatigue life at higher operating 

temperatures. 

AISI M-50 has been used successfully for several years 

as a rolling-element bearing material for temperatures to 

315 “C (600 “F) (refs. 40 to 44). It has also been used for 

T Elongation 

in 5 cm (2 in). 

percent 

---- 

3 

20 

27 

19 

18 

16 

14 

13 

15 

14 

19 

35 

16 

15 

15 

16 

6.8 

16 

19 

lightly loaded accessory gears for aircraft applications at 

high temperatures. However, the standard AISI M-50 

material is generally considered too brittle for more 

heavily loaded gears. AISI M-50 is considerably better as 

a gear material when forged with integral teeth. The grain 

flow from the forging process improves the bending 

strength and impact resistance of the AISI M-50 

considerably (ref. 45). The AISI M-50 material can also 

be ausforged with gear teeth to give good bending 

strength and better pitting life (refs. 46 and 47). 

However, around 760 “C (1400 OF) the ausforging 

temperature is so low that forging gear teeth is difficult 

and expensive. As a result ausforging for gears has had 

considerably limited application (ref. 46). Test results 

show that the forged and ausforged gears can give lives 

approximately three times those of the standard AISI 

9310 gears (ref. 46). 

Nitralloy N is a low-alloy nitriding steel that has been 

used for several years as a gear material. It can be used 

for applications requiring temperatures of 204 to 232 “C 

(400 to 450 “F). A modified Nitralloy N called Super 

Nitralloy or 5Ni-2Al Nitralloy was used in the U.S. 

supersonic aircraft program for gears. It can be used for 

gear applications requiring temperatures to 260 “C 

(500 “F). Table 7 gives relative surface fatigue data for 

Super Nitralloy. 

Two materials that were developed for casecarburized, tapered roller bearings but also show promise 

as high-temperature gear materials are CBS 1OOOM and 

CBS 600 (refs. 48 and 49). These materials are low- to 

medium-alloy steels that can be carburized and hardened 

to give a hard case of Rockwell C-60 with a core of 

TABLE 7.-RELATIVE SURFACE PITTING 

FATIGUE FOR VAR AISI 9310 STEEL AND 

AIRCRAFT-QUALITY GEAR STEELS 

(ROCKWELL 59-62) 

Steels Ten-percent 

relative 

life 

VAR AISI 9310 1 

VAR AlSI 9310 (shot peened) 1.6 

VAR Carpenter EX-53 2.1 

CVM CBS 600 1.4 

VAR CBS 1000 2.1 

CVM VASCO X-2 2.0 

CVM Super Nitralloy (5Ni-2AI) 1.3 

Vl,M-VAR AISl M-50 (forged) 3.2 

VIM-VAR AISI M-50 (ausforged) 2.4 

22 
Rockwell C-38. Surface fatigue test results for CBS 600 

and AISI 93 10 are also shown in table 7. The CBS 600 has 

a medium fracture toughness that can cause fracture 

failures after a surface fatigue spa11 has occurred. 

Two other materials that have recently been developed 

as advanced gear materials are EX-53 and EX-14. 

Reference 50 reports that the fracture toughness of 

EX-53 is excellent at room temperature and improves 

considerably as temperature increases. The EX-53 

surface fatigue results show a lo-percent life that is twice 

that of the AISI 9310. 

Vasco X-2 is a high-temperature gear material (ref. 5 1) 

that is currently being used in advanced CH-47 helicopter 

transmissions. This material has an operating temperature limit of 315 “C (600 OF) and has been shown to 

have good gear load-carrying capacity when properly 

heat treated. The material has a high chromium content 

(4.9 percent) that oxidizes on the surface and can cause 

soft spots when the material is carburized and hardened. 

A special process has been developed that eliminates 

these soft spots when the process is closely followed (ref. 

52). Several groups of Vasco X-2 with different heat 

treatments were surface fatigue tested in the NASA gear 

test facility. All groups except the group with the special 

processing gave poor results (ref. 53). Vasco X-2 has a 

lower fracture toughness than AISI 9310 and is subject to 

tooth fractures after a fatigue spall. 

3.2 Metallurgical Processing Variables 

Research reported in the literature on gear metallurgical processing variables is not as extensive as that for 

rolling-element bearings. However, an element of 

material in a Hertz stress field does not recognize whether 

it is in a bearing or a gear. It only recognizes the resultant 

shearing stress acting on it. Consequently the behavior of 

the material in a gear will be much like that in a rollingelement bearing. The metallurgical processing variables 

to be considered are 

(1) Melting practice, such as air, vacuum induction, 

consumable-electrode vacuum remelt (CVM), vacuum 

degassing, electroslag (electroflux) remelt, and vacuum 

induction melting-vacuum arc remelting (VIM-VAR); 

(2) Heat treatment to give hardness, differential 

hardness, and residual stress 

(3) Metalworking, consisting of thermomechanical 

working and fiber orientation 

These variables can significantly affect gear performance. Other factors that can also significantly affect 

gear fatigue life and that have some meaningful 

documentation are not included. These are trace 

elements, retained austenite, gas content, and inclusion 

type and content. Although any of these factors can 

exercise some effect on gear fatigue life, they are, from a 

practical standpoint, too difficult to measure or control 

by normal quality control procedures. 

Heat treatment procedures and cycles, per se, can 

also affect gear performance. However, at present no 

controls, as such, are being exercised over heat treatment. 

The exact thermal cycle is left to the individual producer 

with the supposition that a certain grain size and hardness 

range be met. Hardness is discussed in some detail in 

reference to gear life. Lack of definitive data precludes 

any meaningful discussion of grain size. 

Melting practice. -Sufficient data and practical 

experience exist to suggest that the use of vacuum-melted 

materials, and specifically CVM, can increase gear 

surface pitting fatigue life beyond that obtainable with 

air-melted materials (refs. 54 to 59). Since, in essentially 

all critical applications such as gears for helicopter 

transmissions and turboprop aircraft, vacuum-melted 

material is specified, a multiplication factor can be 

introduced into the life estimation equation. 

Life improvements over air-melted steels to 13 times by 

CVM processing (refs. 41 to 44) and to 100 times by 

VIM-VAR processing (ref. 59) are indicated in the 

literature. However, it is recommended that conservative 

estimates of life improvements be considered, such as a 

factor of 3 for CVM processing and a factor of 10 for 

VIM-VAR processing. Although these levels may be 

somewhat conservative, the confidence factor for 

achieving this level of improvement is high. Consequently 

the extra margin of reliability desired in critical gear 

applications will be inherent in the life calculations. Data 

available on other melting techniques such as vacuum 

induction, vacuum degassing, and electroslag remelting 

indicate that the life improvement approaches or equals 

that with the CVM process. However, it is also important 

to differentiate between CVM (consumable-electrode 

vacuum melting) and CVD (carbon vacuum degassing). 

The CVM process yields cleaner and more homogeneous 

steels than CVD. 

Heat treatment.-Gears are heat treated by furnace 

hardening, carburizing, induction hardening, or flame 

hardening (ref. 14). Gears are case hardened, through 

hardened, nitrided, or precipitation hardened for the 

proper combination of toughness and tooth hardness. 

Using high-hardness, heat-treated steels permits smaller 

gears for a given load. Also, hardening can significantly 

increase service life without increasing size or weight. But 

the gear must have at least the accuracies associated with 

softer gears and, for maximum service life, even greater 

precision. 

Heat treatment distortion must be minimized for 

longer service life. Several hardening techniques have 

proven useful. For moderate service life increases the 

gears are hardened but kept within the range of 

machinability so that distortion produced by heat 

treating can be machined away. Where maximum 

durability is required, surface hardening is necessary. 

Carburizing, nitriding, and induction hardening are 

generally used. However, precision gearing (quality 10 or 

23 
better) can only be ensured by finishing after hardening. 

Full-contour induction hardening is an economical and 

effective method for surface-hardening gears. The 

extremely high but localized heat allows small sections to 

come to hardening temperatures while the balance of the 

gear dissipates heat. Thus major distortions are 

eliminated. 

Although conventional methods such as flame 

hardening reduce wear by hardening the tooth flank, gear 

strength is not necessarily improved. In fact, stresses built 

up at the juncture of the hard and soft materials may 

actually weaken the tooth. Induction hardening provides 

a hardened tooth contour with a heat-treated core to 

increase both surface durability and tooth strength. The 

uniformly hardened tooth surface extends from the 

flank, around the tooth, to the flank. No stress concentrations are developed to impair gear life. 

Nitriding is a satisfactory method of hardening small 

and medium-size gears. Distortion is minimal because 

furnace temperatures are comparatively low. The hardening pattern is uniform, but the depth of hardness is 

limited. Best results are achieved when special materials 

suited to nitriding are specified. Most gear manufacturing specifications do not designate heat treatment, but 

rather call for material characteristics (i.e., hardness and 

grain size) that are controlled by the heat treatment cycle. 

Hardness is the most influential heat-treatment-induced 

variable (refs. 60 to 62). It is recommended that Rockwell 

C-58 be considered the minimum hardness for critical 

gear applications. 

A relationship has been proposed in reference 42 that 

approximates the effect of hardness on surface fatigue 

life: 

L_z =,o.wc.,-&,I) 

Ll 

where L, and L2 are lo-percent fatigue lives at gear 

hardnesses R~J and Rp2, respectively. Although this 

relationship was obtained for AISI 52100, it can be 

extended to other steels. The life-hardness curve in figure 22 represents equation (48) with the relative life at 

Rockwell C-60 equal to 1 .O. These hardness-temperature 

curves indicate the decrease in hardness with increased 

temperature for various initial room-temperature 

material hardnesses. 

To use the nomograph (fig. 22), first determine the 

gear operating temperature and then follow a horizontal 

line to the appropriate room-temperature hardness curve. 

Next move vertically to the life-hardness curve and read 

the relative life at that point. See the dashed-line example 

for 149 “C (300 “F) operating temperature and Rockwell 

2.0 r 

1.6 

.a 

’ bj16 

Rockwell C hardness at room temperature 

Figure 22.-Nomograph for determining relative life at operating 

temperature as a function of room-temperature hardness. 

C-64 room-temperature hardness. Relative life for this 

example is approximately 1.3. 

Another concept to be considered is the effect of 

differences in hardness between the pinion and the gear 

(refs. 63 and 64). Evidence exists that hardness differences between the mating components can affect system 

life positively by inducing compressive residual stresses 

during operation (ref. 64). Differential hardness m is 

defined as the hardness of the larger of two mating gears 

minus the hardness of the smaller of the two. It appears 

that a mof 2 points Rockwell C may be an optimum for 

maximum life. For critical applications, as a practical 

matter, it would be advisable to match the hardness of 

the mating gears to ensure a m of zero and at the same 

time ensure that the case hardness of the gear teeth at 

room temperature is the maximum attainable. This will 

allow for maximum elevated operating temperature and 

maximum life. The AH effect has been verified 

experimentally for rolling-element bearings, but there is 

no similar published work for gears. 

As previously discussed, residual stresses can be 

induced by the heat-treatment process, differential 

hardness, or shot peening. There is no analytical method 

to predict the amount of residual stress in the subsurface 

region of gear tooth contact. However, these residual 

stresses can be measured in test samples by x-ray 

diffraction methods. The effect of these residual stresses 

24 
on gear pitting fatigue life can be determined by the 

following equation (ref. 37): 

(49) 

where 

rmax maximum shear stress (45” plane) 

s r7Y measured compressive residual stress below 

surface at location of rmax 

Metalworking.-Proper grain flow or fiber orientation 

can significantly extend pitting fatigue life (refs. 65 and 

66) and may improve the bending strength of gear teeth. 

Proper fiber orientation can be defined as grain flow 

parallel to the gear tooth shape. Standard forging of 

gears with integral gear teeth as opposed to machining 

teeth in a forged disk is one way of obtaining proper fiber 

orientation (ref. 46). A controlled-energy-flow forming 

technique (CEFF) can be used for this purpose. This is a 

high-velocity metalworking procedure that has been a 

production process for several years. 

AISI M-50 steel is a through-hardened material often 

used for rolling-element bearings in critical applications. 

Test gears forged from AISI M-50 steel yielded 

approximately five times the fatigue life of machined 

vacuum-arc-remelted (VAR) AISI 9310 gears (ref. 46). 

Despite its excellent fatigue life AISI M-50 is not recommended for gears. Its low fracture toughness makes gears 

prone to sudden catastrophic tooth fracture after a 

surface fatigue spa11 has begun rather than to the gradual 

failure and noisy operation typical of surface pitting. It is 

expected that forged AISI 9310 (VAR) gears would 

achieve similar life improvement while retaining the 

greater reliability of the tougher material. 

Ausforging, a thermomechanical fabrication method, 

has potential for improving the strength and life of gear 

teeth. Rolling-element tests with AISI M-50 steel (not 

recommended for gears-see above) show that 75- to 

80-percent work (reduction of area) produces the 

maximum benefit (refs. 67 to 69). The suitability of 

candidate steels to the ausforging process must be 

individually evaluated. AISI 9310 is not suitable because 

of its austenite-to-martensite transformation characteristic. Tests reported in reference 46 found no statistically 

significant difference in lives of ausforged and standard 

forged AISI M-50 gears. The lack of improvement in the 

ausforged gears is attributed to the final machining 

required, which removes some material with preferential 

grain flow. Reference 46 also reports a slightly greater 

tendency to tooth fracture in ausforged gears. This 

tendency is attributed to poorer grain flow than in 

standard forged gears. The energy required limits the 

ausforging process to gears no larger than 90 mm (3.5 in) 

in diameter. 

3.3 Manufacturing 

Gears can be formed by various processes that can be 

classified under the general headings of milling, generating, and molding. 

Milling.-Almost any tooth form can be milled. 

However, only spur, helical, and straight-bevel gears are 

usually milled. The surface finish can be held to 3.18 pm 

(125 pin) rms. 

Generating.-In the generating process teeth are 

formed in a series of passes by a generating tool shaped 

somewhat like a mating gear. Either hobs or shapers can 

be used. Hobs resemble worms that have cutting edges 

ground into their teeth. Hobbing can produce almost any 

external tooth form except bevel gear teeth, which are 

generated with face-mill cutters, side-mill cutters, and 

reciprocating tools. Hobbing closely controls tooth 

spacing, lead, and profile. Surface finishes as fine as 

1.6 pm (63 pin) rms can be obtained. Shapers are reciprocating pinion- or rack-shaped tools. They can produce 

external and internal spur, helical, herringbone, and face 

gears. Shaping is limited in the length of cut it can 

produce. Surface finishes as fine as 1.6 pm (63 pin) rms 

are possible. 

Molding.-Large-volume production of gears can 

often be achieved by molding. Injection molding is used 

for light gears of thermoplastic material. Die-casting is a 

similar process using molten metal. Zinc, brass, 

aluminum, and magnesium gears are die-cast. Sintering is 

used in making small, heavy-duty gears for instruments 

and pumps; iron and brass are the materials most used. 

Investment casting and shell molding are used for 

medium-duty iron and steel gears for rough applications. 

Gear finishing. -Shaving gears improves accuracy and 

finish while removing only a small amount of surface 

metal. A very hard mating gear with many small cutting 

edges is run with the gear to be shaved. The surface finish 

can be as fine as 0.81 pm (32 pin) rms. 

Lapping corrects minute heat treatment distortion 

errors in hardened gears. The gear is run in mesh with a 

gear-shaped lapping tool or another mating gear. An 

abrasive lapping compound is used between them. 

Lapping improves tooth contact but does not increase the 

accuracy of the gear. Finish is of the order of 0.81 pm (32 

pin) rms. 

25 
Grinding is the most accurate tooth-finishing process. 

Profiles can be controlled or altered to improve tooth 

contact. For example, barreling or crowning the flanks of 

teeth promotes good contact at the center, where the 

tooth is strong, and minimizes contact at the edge and 

corner, where the tooth is unsupported. Surface finishes 

as fine as 0.41 pm (16 pin) rms or better can be obtained. 

However, at surface finishes better than 0.41 pm (16 pin) 

rms, tooth errors can be induced that may affect gear 

tooth dynamic loading and as a result nullify the 

advantage of the improved surface finish. 

4.0 Stresses and Deflections 

There are several approaches to determining the 

stresses and deflections in gears. The one used most 

commonly for determining the bending stress and 

deflection is a modified form of the Lewis equation 

(ref. 8). This approach considers the gear tooth to be a 

cantilevered beam or plate. Modifying factors based on 

geometry or type of application are then used to amend 

the calculated stress and allowed design strength. It is 

now known that the calculated stress may not accurately 

represent the true stress. But since a large amount of 

experience has accrued across the gearing industry, data 

on compatible “allowed stress” have been compiled, and 

the modified Lewis equation is used successfully to design 

gears. The Lewis method should be thought of as a means 

of comparing a proposed new gear design for a given 

application with successful operation of similar designs. 

If the computed stress is less than the computed allowable 

stress, the design will be satisfactory. 

The American Gear Manufacturers Association has 

published a standard (AGMA 218.01) (ref. 70) for 

calculating the bending strength of spur and helical gears, 

as well as many other standards for gear design and 

applications. A second method of calculating stresses and 

deflections comes from classical theory on elasticity. 

Several examples of such methods are found in the 

literature (refs. 71 to 73). The methods involve the use of 

complex variable theory to map the shape of the gear 

tooth onto a semi-infinite space in which the stress 

equations are solved. The methods have the advantage of 

computing accurate stresses in the regions of stress 

concentration. Although for practical use they require a 

computer, the computation times are fast enough that the 

methods can be used in everyday design. They do, 

however, seem limited to plane stress problems. 

Therefore they will not work for bevel or helical gears, 

where the effects of the wide teeth and distributed tooth 

load are important. 

The most powerful method for determining accurate 

stress and deflection information is the finite-element 

method. However, the method is too expensive and 

cumbersome to use in an everyday design situation. 

26 

Although at first the method would seem to answer all 

problems of computational accuracy, it does not. 

Research is continuing in this promising area (refs. 74 to 

79). The versatility of the finite-element method 

introduces other questions of how best to use the method. 

These questions include (1) the number of elements and 

the most efficient arrangement of elements in the regions 

of stress concentrations; (2) how to represent the 

boundary support conditions; and (3) how to choose the 

aspect ratios of the solid brick element in threedimensional stress problems. Pre- and postprocessors 

specifically for gears have been developed (ref. 79). 

Various studies have been conducted around specific 

questions such as the effect of rim thickness on critical 

stress in lightweight gears (refs. 80 and 81). It is expected 

that methods such as the finite-element method and the 

classical theory of elasticity will continue to be used in a 

research mode, with research results being used to supply 

application factors and stress-modifying factors for the 

modified Lewis (AGMA) method. One such approach 

has been suggested for the effect of the ratio of rim 

thickness to gear tooth height (ref. 82). 

Experimental methods and testing of all proposed gear 

applications cannot be overemphasized. The foregoing 

discussion has centered around the bending fatigue 

strength of the gear teeth, where the failure-causing stress 

is at the gear tooth fillet between the tooth profile and the 

root of the gear. Surface pitting fatigue and scoring 

failures are discussed in another section and are also 

likely modes of failure. Experimental testing of gear 

prototype designs will reveal any weakness in the design 

that could not be anticipated by conventional design 

equations, such as where special geometry factors will 

cause the failure-causing stress to appear at locations 

other than the tooth fillet.5 In critical applications such as 

aircraft, full-scale testing of every prototype gearbox is 

done in ground-based test rigs, and every piece of the 

flight-hardware gearbox assembly is tested in a “green 

run” transmission test rig as a distinct step in the 

production process for the completed assembly. An 

experimental verification of a proposed gear design is 

essential to a complete design process. 

4.1 Lewis Equation Approach for Bending Stress 

Number (Modified by AGMA) 

For a gear tooth loaded by the transmitted tangential 

load W,, which is assumed to act at the tip of the tooth 

(fig. 23), the Lewis formula for the stress at the fillet 

5Ref. 82 shows that for thin-rimmed internal gears the maximum 

tensile stress occurs several teeth away from the loaded tooth. Ref. 79 

shows that for thin-rimmed external gears the bending stress in the root 

can be higher than the fillet stress. These observations emphasize the 

need for care in applying conventional design equations with a cavalier 

confidence in their universal applicability. 
Maximum 

stress point, sinscribed 

parabola - 

Figure 23.-tiear tooth load. 

follows the simple strength-of-materials formula for 

bending stress in a beam: 

s _ kfc _ (WJW _ 6W& 

b I ~ Ft3/12 Ft= (50) 

The values for the tooth thickness at the critical section 

and the beam length are obtained from inscribing a 

parabola inside the tooth, tangent to the fillet, with the 

apex at the point of load application. The procedure is 

based a parabolic-shaped beam being a beam of constant 

bending stress. 

For a family of geometrically similar tooth forms a 

dimensionless tooth form factor Y is defined as follows: 

r2D 

y=!g (51) 

where P is the diametral pitch. From the form factor the 

tooth bending stress is calculated by 

sb= (9) (+) 

where 

W, transmitted tangential load 

F face width 

(52) 

The assumptions for which the preceding equation was 

derived are as follows: 

(1) The radially directed load was neglected. 

(2) Only one tooth carried the full transmitted load. 

(3) There was uniform line contact between teeth, 

causing a uniform load distribution across the face width. 

(4) The effect of stress concentration at the fillet was 

neglected. 

A more comprehensive equation has been defined by 

AGMA as follows: 

sb= (9) ($)(Ka$Km) 

where 

sb bending stress, MPa (psi) 

K, application factor 

KS size factor 

K,,, load distribution factor 

KV dynamic load factor or velocity factor 

J geometry factor 

(53) 

The application factor K, is intended to modify the 

calculated stress according to the type of service the gear 

will see. Some of the pertinent application influences 

include type of load, type of prime mover, acceleration 

rates, vibration, shock, and momentary overloads. 

Application factors are established after considerable 

field experience has been gained with a particular type of 

application. The designer should establish application 

factors based on past experience with actual designs that 

have run successfully. If the designer is unable to do this, 

suggested factors from experience (table 8) may be used. 

The size factor KS reflects the influence of nonhomogeneous materials. From the weakest-link theory 

we can expect a large section of material to be weaker 

than a small section because of the greater probability of 

the presence of a “weak” link. The size factor KS 

corrects the stress calculation to account for the known 

fact that larger parts are more prone to fail. At present, a 

size factor of KS= 1 is recommended for spur and helical 

gears by the AGMA; for bevel gears The Gleason Works 

recommends 

TABLE 8.-SUGGESTED APPLICATION 

FACTORS K, FOR SPEED-INCREASING 

AND -DECREASING DRIVES’ 

Power 

source 

jniform 

Light shock 

bledium shock 

Load on driven machine 

Uniform Moderate Heavy 

shock shock 

KU 

1 .oo I .25 21.75 

1.25 1.50 22.00 

1.50 1.75 ~2.25 

27 
~~~~-0.25 P< 16 

Ks=0.5 P>16 

(54) 

The load distribution factor Km is intended to account 

for the effects of possible misalignments in the gear that 

will cause uneven loading and a magnification of the 

stress above the uniform case. If possible, the ratio of 

face width to pitch diameter should be kept small (<2) 

for less sensitivity to misalignment and uneven load 

distribution due to load-sensitive deflections. A second 

rule of thumb is to keep the face width between three and 

four times the circumferential pitch. Often gear teeth are 

crowned or edge relieved to diminish the effect of 

misaligned axes on tooth stresses. Normally the range of 

this factor is approximately 1 to 2. A factor larger than 2 

should be used for gears mounted with less than full face 

contact. When more precision is desired and enough 

detail of the proposed design is in hand to make the 

necessary calculations, consult the AGMA standards. 

For most preliminary design calculations use the data in 

table 9 for spur and helical gears and those in figure 24 

and table 10 for bevel gears. 

The dynamic load factor Ky is intended to correct for 

the effects of the speed of rotation and the degree of 

precision on gear accuracy. A first approximation for Kv 

can be taken from the chart in figure 25. This chart is 

intended only to account for the effect of tooth 

inaccuracies. It should not be used for lightly loaded, 

lightly damped gears or for resonant conditions. If the 

gear system approaches torsional resonance, or if the 

gear blank is near resonance, computerized numerical 

methods such as those presented in references 83 and 84 

must be used. The chart in figure 26 shows the reciprocal 

of the dynamic factor as a function of speed for a certain 

TABLE 9.-LOAD DISTRIBUTION FACTORS K,, FOR SPUR AND HELICAL GEARS 

Condition of support Face width, F, cm (in) 

15 (2) 15 (6) 23 (9) 31 (16) 

Spur Helical Spur Helical Spur Helical Spur Helical 

Accurate mounting, low bearing 1.3 1.2 1.4 1.3 1.5 1.4 1.8 1.7 

clearances, minimum elastic 

deflection, precision gears 

Less rigid mountings, less 

accurate gears, contact across 

full face 

1.6 1.5 1.7 1.6 1.8 1.7 2.0 2.0 

Accurate and mounting such 

that less than full face 

contact exists 

Over 2.0 

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 

Face width, 

F, 

cm (in) 

Load distribution factor, Km 

Figure 24.-Load distribution factor for bevel gears. 

28 
TABLE IO.-LOAD DISTRIBUTION FACTORS K,,, FOR BEVEL GEARS 

Application Both members One member Neither member 

straddle mounted straddle mounted straddle mounted 

K?l 

I I I 

General industrial 1.00 to 1.10 1.10 to 1.25 1.25 to 1.40 

Automotive 1.00 to 1.10 1.10 to 1.25 ---------- 

Aircraft 1.00 to 1.25 1.10 to 1.40 1.25 to 1.50 

accuracy 

k- Very accurate gearin; level 

4 I1 

z P 2 10 

.v 8 9 

Fij 7 

5 0 

0 5 10 15 20 25 30 35 413 45 50 

Pitch-line velocity. vt, mfs 

Pitch-line velocity, vt, ftlmin 

Figure 25.-Dynamic load factor. (Transmission accuracy level number 

specifies the AGMA class for accuracy of manufacture.) 

Critical damping, 

percent 

h 

ii 1.6 

; 

2 

z 

1.2 

27 

‘i; 

5 .a 

iz .cZ 

kz c& .4 

0 10 20 30 40 50 60 70 

Pitch-line velocity, q, m/s 

0 2 4 6 8 10 12 14x103 

Pitch-line velocity, vt, ftlmin 

Figure 26.-Dynamic load factor as a function of damping ratio. Gear 

ratio, mo, 1; number of teeth, N, 28; diametral pitch, P, 8. 

specific design for three values of damping. No 

comprehensive data for a broad range of parameters are 

yet available. However, the computer programs TELSGE 

(Thermal Elastohydrodynamic Lubrication of Spur 

Gears) and GRDYNSING (Gear Dynamic Analysis for 

Single Mesh) are available from the NASA computer 

program library.6 

6Available from the Computer Software Management and 

Information Center (COSMIC), Computing and Information Services, 

112 Barrow Hall, The University of Georgia, Athens, Georgia 30602. 

The geometry factor J is a modification of the form 

factor Y to account for three more influences: stress 

concentration, load sharing between the teeth, and 

changing the load application point to the highest point 

of single-tooth contact. The-form factor Y can be found 

from a geometrical layout of the Lewis parabola within 

the gear tooth outline (fig. 23). The load-sharing ratio 

can be worked out by using a statistically indeterminate 

analysis of the pairs of teeth in contact that considers 

their flexibilities. Stress concentration factors from the 

work of Dolan and Broghammer (ref. 85) can be applied 

to get J. The J factors have already been computed for a 

wide range of standard gears and are available in the 

AGMA standards. For many gear applications the charts 

presented in figures 27 to 32 for spur, helical, and bevel 

gears can be used. 

The transmitted tooth load W, is equal to the torque 

divided by the pitch radius for spur and helical gears. For 

bevel gears the calculation should use the large-end pitch 

cone radius. 

A rim thickness factor Kb that multiplies the stress 

computed by the AGMA formula (eq. (53)) has been 

proposed in reference 82. The results in figure 33 show 

that for thin-rimmed gears the calculated stress should be 

adjusted by a factor Kb if the backup ratio is less than 2. 

For spiral-bevel gears a stress correction factor has been 

Addendum, 

1.000 

Generating rack, 1 pitch 

Pinion addendum, 1.030 

Gear addendum, 1. ,000 

Number of teeth 

in mating gear 

rlb30 

- /r 170 

/If- 55 

‘-12 15 17 20 24 30 40 60 125 275~ 

Number of teeth for which geometry factor is desired 

Figure 27.--Geometry factor for 20” spur gear. Standard addendum. 

29 
rAdd?ndum, 

//’ 1.033 Pinion addendum, 1.000 

Gear addendum, 1.000 

Generating rack, 1 pitch 

.7 

r 1000 

Load applied at highest point ’ 

of single-tooth contact 

.2 

12 15 17 20 24 30 40506080 27500 

Number of teeth for which geometry factor is desired .16 .20 .24 .28 

Geometry factor, J 

.32 .M 

Figure 28.-Geometry factor for 25” spur gear. Standard addendum 

Figure 31.-Geometry factor for spiral-bevel gears with 20” pressure 

angle, 35” spiral angle, and 90” shaft angle. 

Number 

of teeth 

5D3 

.4 

5 10 15 20 25 30 35 

Helix angle,*, deg 

Figure 29.-Geometry factor for 20” normal-pressure-angle helical 

gear. Standard addendum; full fillet hob. 

Number of teeth in 

1.05 - mating element 

500 

150 - 

75 

s’ 1.00 

25 50 

m 

#:;;;, 

5 10 15 20 25 30 35 

Helix angle.*, deg 

Figure 30.-Geometry factor multipliers for 20” normal-pressure-angle 

helical gear. (Modifying factor can be applied to J factor when other 

than 75 teeth are used in mating element.) 

Number of teeth in 

mating element 

13 20 30 40 50 

.24 .2a .32 .35 .40 

Geometry factor, J 

Figure 32.--Geometry factor for straight bevel gears with 20” pressure 

angle and 90” shaft angle. 

30 
2. 6 r JTr’--R-, i a Photoelastic test data 

0 Finite-element-method 

analysis data 

0 Data from experiment 

$ 1.8 

5 

.z 

2 1.4 

.- 

w 

1. 0 

.4 .6 .8 1 2 34 6 8 10 

Backup ratio, rim thickness/tooth height 

Figure 33.-Rim thickness factor 

given by Gleason to account for the effects of cutter 

radius curvature, which is a measure of the tooth 

lengthwise curvature. The Gleason equation for cutter 

radius factor is 

0.2788 

log sin I+!J 

+ 0.7889 

where 

A gear mean cone distance 

$ gear mean spiral angle 

TC cutter radius 

(55) 

Normally t-J.4 ranges from 0.2 to 1 and $ from 10” to 

50”, giving a KX range of 1 to 1.14. Recommended 

practice is to select the next larger nominal cutter that has 

a radius given by 

rc= 1.1 A sin tc/ (56) 

The cutter radius factor should be entered in the 

denominator of the stress equation (eq. (53)). 

4.2 Allowable Bending Stress Number 

The previous section has presented a consistent method 

for calculating a bending stress index number. The 

stresses calculated by equation (53) may be much less 

than stresses measured by strain gauges or calculated by 

other methods such as finite-element methods. However, 

there is a large body of allowable stress data available in 

the AGMA standards that is consistent with the 

calculation procedure of equation (53). If the calculated 

stress S is less than the allowable tensile stress S,,, the 

design should be satisfactory. The equation is 

(57) 

where the K factors modify the allowable stress to 

account for the effects that alter the basic allowed stress 

in bending. Table 11 shows the basic allowable stress in 

bending fatigue for lo7 stress cycles for single-direction 

bending. Use 70 percent of these values for reverse 

bending such as in an idler gear. If there are momentary 

overloads, the design should be checked against the 

possibility of exceeding the yield strength and causing 

permanent deformation of the gear teeth. For throughhardened steel the allowable yield strength is dependent 

on Brine11 hardness HB according to the following 

empirical equation: 

&, = (482 HB - 69 kPa) 

= (482 HB - 32 800 psi) (58) 

The life factor KL can be selected from table 12 and is 

used to adjust the allowable stress for the effect of 

designing to lives other than 107 stress cycles. 

The temperature factor KT is used to derate the 

strength in order to account for loss of basic material 

strength at high temperature. For temperatures to 121 “C 

(250 OF) the factor is unity. For higher temperatures 

strength is lost from the tempering effect, and in some 

materials at over 177 “C (350 “F), KT must be greater 

than unity. One alternative method that is used widely is 

to form the KT factor 

K = 46O+“C 460+“F 

T-=- 620 620 (59) 

where the temperature range is between 71 and 149 “C 

(160 and 300 “F). Operation at temperatures above 

149 “C (300 OF) is normally not recommended. 

The reliability factor KR is used to adjust for desired 

reliability levels either less or greater than 99 percent, 

which is the level for the allowable bending strength data 

in table 11. If statistical data on the strength distribution 

of the gear material are in hand, a suitable KR factor can 

be selected. In lieu of this, use the values from table 13. 

4.3 Other Methods 

There is no available methodology that applies to all 

gear types for stress and deflection analysis. However, 

some selected methods that have a limited application are 

practical for design use and for gaining some insight into 

the behavior of gears under load. 

31 
TABLE 11 .-ALLOWABLE BENDING STRESS NUMBER S,,, FOR 10’ STRESS CYCLES 

1 AGMA Commercial Heat 

class designation treatment 

Core Spur and 

helical 

gear 

Bevel 

Through 

hardened 

and 

rempered 

Flame- or 

inductionhardened 

section 

Flame- or 

inductionhardened 

section 

Carburized 

and case 

hardened 

Nitrided 

I 

As cast 

As cast 

As cast 

Minimum 

hardness 

surface 

MPa ksi MPa 

180 BHN 

240 BHN 

300 BHN 

360 BHN 

400 BHN 

50-54 HRC 

------ 

------ 

173-228 25-33 

210-280 31-41 

250-320 36-47 

280-360 $0-52 

290-390 12-56 

310-380 $5..55 

55 HRC (min) _----- _---- 207 30 

150 22 

55 HRC 

60 HRC _----- 

380-450 55-65 190 27.5 

380-480 55-70 207 30 

48 HRC !OO BHN 230-310 34-45 

46 HRC KM BHN 250-325 16-47 

60 HRC IO0 BHN 260-330 !8-48 

54-60 HRC 150 BHN 380-450 i5-65 

Material 

Steel A-l to A-5 ASTM 100-70-03 

41SI 4140 

41SI 4340 

Vitralloy 135M 

1.5 Percent 

:hromium 

Zast iron 20 

30 

40 

- 

ksi 

- 

14 

19 

_--_ 

13.5 

20 

_--_ 

2.7 

4.6 

7.0 

- 

8.0 

11 

14 

18.5 

- 

97 

131 

93 

140 

35 5 20 

60 8.7 32 

90 13 50 

175 BHN 

200 BHN 

Uodular 

ductile) 

r0n 

A-7-a 

A-7-c 

A-7-d 

A-7-e 

GTM 60-40- 18 

\ST,M 80-55-06 

104 15 55 

140 20 76 

180 26 

207 30 

97 

130 

Annealed 

Quenched and 

tempered 

Quenched and 

tempered 

Quenched and 

tempered 

140 BHN 

180 BHN 

230 BHN 

270 BHN 

165 BHN 

180 BHN 

195 BHN 

240 BHN 

rensile 

strength (min), 

275 MPa 

(40 000 psi) 

rensile 

strength (min), 

620 MPa 

(90 000 psi) 

------ 

------ 

_----- 

_----- 

\STM 100-70-03 

XSTM 120-90-02 

vlalleable 

ron 

pearlitic) 

A-8-c 

A-8-e 

A-8-t 

A-8-i 

15007 

iooo5 

i3007 

10002 

10 

13 

16 

21 

163 23.6 

3ronze Bronze 2 

4l/Br, 

\GMA 2C Sand cast 

GTM B-148-52 Heat treated 

(alloy 9C) 

32 
TABLE 12.-LIFE FACTOR K, 

Number 

of 

cycles 

Spur, helical, and herringbone gears Bevel gears 

(case car160 250 450 Case burizeda) 

BHN BHN BHN carburizeda 

4 

Up to 1000 1.6 2.4 3.4 2.7 4.6 

10 000 1.4 1.9 2.4 2.0 3.1 

100 000 1.2 1.4 1.7 1.5 2.1 

1 million 1.1 1.1 1.2 1.1 1.4 

10 million 1.0 1.0 1.0 1.0 1.0 

100 million 1 .O-0.8 1 .O-0.8 1.0-0.8 1.0-0.8 4.6 

and over 

aRockwell C-55 (mm). 

TABLE 13.-RELIABILITY FACTOR KR 

I Requirements of application 1 KR 1 Reliability 1 

A method for determining the stress in bending has 

been successfully applied to spur gears (ref. 86). The 

stress is calculated by the following equation: 

s = wNcosQ~ 

t F [I +0.16(k)s’7] [ 3 + (31’2 

X tL l+TYtanrpi 

) I 

tan QL 

- - 

ts (60) 

s 

where 

St root fillet tensile stress at location indicated in 

fig. 34 

F tooth face width measured parallel to gear axis 

WN instantaneous force that is normal to tooth surface 

and transmitted by tooth 

V Poisson’s ratio 

and the remaining quantities in equation (60) are defined 

in figure 34. Angle y in figure 34 defines the area where 

the root fillet tensile stress is calculated by equation (60). 

Cornell (ref. 86) provides an equation for the value of y 

that locates the position of the maximum root stress: 

(1 +0.16Ap.7)Ai 

tan yi+ I= 

Bi(4+0.416AP”) - (5 +0.0f6ap.7)atanQi 

(61) 

I 4 h \ 

,-Base 

_---- ----___ / ’ circle 

Figure 34.-Tooth geometry for evaluation of root bending stress 

formula parameters. 

where 

Flj= ‘;” +2(1- cosyJ (62) 

and 

Bi= p + sinyi (63) 

where to and 10 are defined in figure 34 and subscripts i 

and i + 1 denote that the transcendental equation (61) can 

be solved iteratively for y, with i standing for the steps in 

the iteration procedure. Once the angle y is determined, 

the dimension t, shown in figure 34 also is determined by 

the formula 

t,=t,+2r(l- cos y) w 

Deflections of gear teeth at the load contact point are 

important in determining the correct amount of tip relief 

to minimize dynamic loads. A finite-element study has 

been done for a gear with a fully backed tooth (thick rim) 

(ref. 84). Figure 35 shows the results. The ordinate is the 

dimensionless deflection 6EF/ W, where 6 is deflection, 

WN/Fis the load per unit of face width, and E is Young’s 

modulus. The dimensionless deflection depends only on 

the load position and the tooth/gear blank configuration 

and not on the size of the gear. The loading positions are 

numbered 1 to 10 with the pitch point being loading 

33 
36 Normal 

rload, 

\ WN 

32 

I28 

8- 

\ \ 

Loading 

position 

l 2 

4 

f-k 

6 

8 

9 

4 

l- 10 

0 10 20 30 43 50 60 7C 

Number of teeth, N 

Figure 35.-Dimensionless deflection as a function of number of gear 

teeth and loading position. 

position 5.5. These deflection values are also useful in the 

construction of a mathematical model of torsional 

vibration in the gear train. 

The methods presented in reference 86 allow study of 

the separate effects contributing to gear tooth deflection. 

The method is based on beam deflection theory with 

corrections included to account for Hertz deflection, 

1‘4 15 - 

10 - 

E, 

- 2 L= 

fii 

F 

-a 5- 

- a- 

-10 

shear deformation, and the flexibility effects of the gear 

blank itself. The results of the method applied to a 

8.9-cm (3.5-in) pitch diameter gear with 28 teeth, a 

diametral pitch of 8, a face width of 0.64 cm (0.25 in), 

and a solid gear blank are shown in figure 36. This figure 

gives an idea of the relative contributions of individual 

flexibility effects to the total deflection. The Hertz 

deflection is approximately 25 percent of the total. The 

gear blank deflection is approximately 40 percent. The 

beam bending effect of the tooth is approximately 35 

percent of the total. Beam bending affects three regions: 

the fillet; the region of blending between the fillet and the 

involute profile (labeled “undercut” in fig. 36); and the 

tooth proper (labeled “involute” in fig. 36). Figure 36 

shows deflection as a function of load position on the 

tooth. The discontinuities in slope are the points where 

the load sharing changes from single tooth-pair contact 

to double tooth-pair contact. 

5.0 Gear Life Predictions 

Gears in operation are subjected to repeated cycles of 

contact stress. Even if the gears are properly designed to 

have acceptable bending stresses for the avoidance of 

bending fatigue failure, there remains the likelihood of 

failure by surface contact fatigue. Contact fatigue is 

unlike bending fatigue in that there is no endurance limit 

for contact fatigue failure. Eventually a pit will form 

when the gear has been stressed by enough repeated load 

cycles. This section shows how to use the method that 

was developed at NASA for predicting the surface fatigue 

life of gears (refs. 87 to 91). The method is based on the 

assumption that pitting fatigue failures for gears and 

m Hertz 

n involute 

m Undercut 

Position on line of action, mm 

-. 3 -. 2 -, 1 0 .l .2 .3 

Position on line of action, in 

Figure 36.-Tooth deflection calculated by method of Cornell 

34 
bearings are similar. The theoretical basis of the life 

prediction formulas is esplained briefly in this section. 

5.1 Theory of Gear Tooth Life 

The fatigue life model proposed in 1947 by Lundberg 

and Palmgren (ref. 92) is the commonly accepted theory 

for the determination of the fatigue life of rollingelement bearings. The probability of survival as a 

function of stress cycles is expressed as 

(65) 

Hence, if the probability of survival S is specified, the life 

7 in millions of stress cycles for the required reliability S 

can be considered a function of the stressed volume V, 

the maximum critical shear stress TV, and the depth to the 

critical shear stress z. As a result, the proportionality can 

be written as 

(66) 

This formula reflects the idea that greater stress shortens 

life. The depth below the surface z at which the critical 

shear stress occurs is also a factor. A microcrack 

beginning at a point below the surface requires time to 

work its way to the surface. Therefore we expect that life 

varies by the power of z. The stressed volume I/is also an 

important factor. Pitting begins near any small stressraising imperfection in the material. The larger the 

stressed volume, the greater will be the likelihood of 

fatigue failure. By the very nature of the fatigue failure 

phenomenon, it is the repetitive application of stress that 

causes cumulative damage to the material. The greater 

the number of stress cycles, the greater will be the 

probability of failure. Experience shows that the failure 

distribution follows the Weibull model, with e = 2.5 for 

gears. Experiments have determined that the exponents c 

and h equal 23.25 and 2.75, respectively. The following 

formulas, based on direct application of statistical 

fundamentals and the Hertz stress equations, enable the 

straightforward calculation of fatigue life for spur gears. 

The expression for reliability as a function of stress cycles 

on a single gear tooth is 

log;=(~)elof3 (&J (67) 

s probability of survival 

T,, life corresponding to a go-percent reliability, 

millions of stress cycles 

The dispersion or scatter parameter e (Weibull exponent) 

based on NASA fatigue tests is equal to 2.5. The singletooth life T,, is determined from 

where Cl0 is 6.44 x lo9 for newtons and millimeters or 

9.18 x 10’8 for pounds and inches. 

W, normal load, N (lb) 

Fe loaded or effective face width, mm (in) 

I loaded profile length, mm (in) 

CP curvature sum, mm - l (in - l) 

The curvature sum is given by 

where 

T~,~, rP,2 pitch circle radii, m(in) 

(a pressure angle, deg 

(6% 

The factor 9.18 x 10’s is an experimentally obtained 

constant based on NASA gear fatigue tests of aircraftquality gears made of AISI 9310 (VAR) steel. If another 

material with a known fatigue life expectancy relative to 

the AISI 9310 material is selected, the life calculated by 

equation (68) for T,o should be multiplied by an 

appropriate factor. The relative lives for several steels 

based on NASA gear fatigue data (table 7) can be used as 

life adjustment factors. A similar treatment of life 

adjustment factors was made for rolling-element bearings 

for the effects of material, processing, and lubrication 

variables (ref. 93). As is always true, the designer should 

base his/her life calculations on experimental life data 

whenever possible. An alternative procedure for 

estimating life is given by the American Gear 

Manufacturers Association (AGMA) standard 218.01 

(ref. 70). The method is based on allowable stress, which 

may be corrected for certain application and geometry 

effects. Because standard 218.01 is widely available, the 

method will not be repeated here. The AGMA method, 

unlike the Lundberg and Palmgren (ref. 92) approach 

presented herein, includes neither the stressed volume 

directly nor the depth to critical stress sensitivities. 

Example problem.-A gear is to be made of an 

experimental steel that has a known life three times that 

of AISI 9310 (based on experimental surface fatigue 

results for a group of specimens). The gear has a pressure 

angle of 20”, a pitch of 8 (module of 3.18 mm/tooth), 28 

teeth, a face width of 9.52 mm (0.375 in), and a standard 

addendum. The gear is to mesh with an identical gear. 

Determine the expected life at the 90- and 99-percent 

35 
reliability levels for a single tooth if the mesh is to carry 

149 kW (200 hp) at 10 000 rpm. 

Solution.-First, the necessary geometry and loading 

must be worked out. The pitch radius from equations (1) 

and (5) is 

1 rp,l = rp,2 = 2 (28 teeth)(3.18 mm/tooth) = 44.5 mm 

1 28 teeth rp,] = rp,2 = - x 2 8 teeth/inch of pitch diameter = 1.75 in 

Also, the base circle radius is 

rb,l = rb,2 = rp cm cp = (44.5 mm) cos (20”) 

=41.8 mm (1.646 in) 

The load normal on the gear tooth surface is determined 

from the power. The torque is 

T= 149kW 

10 000 rpm 

x $ x & = 142.3 N-m (1259 lb-in) 

The normal load is 

w,= T 

~ = L = l;;‘;;; = 3.40 kN (765 lb) rpcosCp rb . 

The standard addendum gear tooth has an addendum 

radius of r, = rp + 1 /P (note that 1 /P = module). Therefore 

the addendum radius is 47.6 mm (1.875 in). The contact 

ratio is defined as the average number of tooth pairs in 

contact. It can be calculated by dividing the contact path 

length by the base pitch. The length of the contact path 

along the line of action from equation (8) is 

Z=2[(47.6mm)2-(41.8mm)2]“2-2(44.5mm) sin (20”) 

= 15.10 mm (0.595 in) 

The base pitch is 

Pb=2T($) =za( ‘F) =9.38 mm (0.369 in) 

Therefore, the contact ratio is 1.61, which means that the 

gear mesh alternates between having one pair of teeth and 

two pairs of teeth in contact at any instant. Thus the load 

is shared for a portion of the time. Since the loading is 

most severe with a single pair of teeth in contact, the 

loaded profile length corresponds to the profile length for 

a single tooth-pair contact. If the contact ratio is greater 

than 2.0, two separate zones of double tooth-pair contact 

contribute to the loaded profile length. For the current 

example with identical gears in mesh, the pinion roll 

angle from the base of the involute to where the load 

begins (lowest point of double tooth-pair contact) is 

EC (~LJ,I + rp,2) sin cp - (2 2 - 4 , 2)l/2 

‘b,l 

tc = 

(44.5 mm + 44.5 mm) sin 20” - [(47.6mmq - (41.8 mmT]“2 

41.8mm 

= 0.1835 rad 

The pinion roll angle increment for the double tooth-pair 

contact (light load zone) is 

2 -Pb lS.lOmm-9.38mm 

CL= __ = 41.8mm =0.1368 rad 

‘b,l 

The pinion roll angle increment for single tooth-pair 

contact (heavy load zone) is 

2pb-2 

fH= ~ = 

2(9.38mm)- I5.10mm =. 0876 rad 

rb,l 41.8 mm 

The length of involute corresponding to the single toothpair contact is 

ll=rb,leH(‘C+CLf ?f) 

II =(41.8mm)(0.0876) 0.0876 0.1835+0.1368-t 2 

> 

II = 1.333 mm (0.0525 in) 

The curvature sum from equation (69) is 

c ( 1 1 

> 

1 

‘= 44.5mm + 44.5 mm sin) 

=0.1314 mm-l (3.34 in-l) 

Finally the tooth life is obtained as follows from equation 

(68): 

Tlo = (6.44 x 109)(3400)-4.3(9.52)3.9 

=627 million stress cycles 

36 
In hours, the life is 

Tlo = (627 x lo6 cycles) 

’ ( 10 OOOrpm l >w 

= 1045 hr 

Conversion to 99-percent reliability by using equation 

(67) yields 

l/2.5 

T1 = (1045 hr) = 408 hr 

5.2 Gear Life 

From basic probability notions the gear life is based on 

the survival probabilities of the individual teeth. For a 

simple gear with N teeth meshing with another gear the 

gear life corresponding to the tooth life at the same 

reliability level is given by the following relation: 

Glu = TlcN- l/e (simple gear) (70) 

where 

e Weibull slope (2.5) 

Grc lo-percent gear life 

Tlo lo-percent single-tooth life as determined by 

previous equations 

N number of teeth 

When the power is transferred from an input gear to an 

output gear through an idler gear, both sides of the tooth 

are loaded once during each revolution of the idler gear. 

This is equivalent to a simple gear with twice the number 

of teeth. Therefore the life of an idler gear with Nteeth is 

approximately 

Grc = Tlo(2ZV)- l/e (idler gear) (71) 

when it meshes with gears of equal face width. The life of 

a power-collecting (bull) gear with N teeth where there 

are u identical pinions transmitting equal power to the 

bull gear is 

i-10 Glc= -(N)-l’e(bullgear) 

u (72) 

5.3 Gear System Life 

From probability theory it can be shown that the life of 

two gears in mesh is given by 

L=(Lie+LFe)-l’e (73) 

where L1 and L2 are the gear lives. This equation can be 

generalized to n gears, in which case 

/ nl \-l/e 

L= (gL7) (74) 

If the power transmission system has rolling-element 

bearings, the system life equations must be solved with 

different values for the Weibull scatter parameter or 

slope. For cylindrical or tapered bearings e = 3/2 and for 

ball bearings e= 1019. The probability distribution for 

the total power transmission is obtained from the 

following equation (95) (ref. 94). 

log(&) = log(&) [ ($el+ (&Jez 

+ (6)‘3+. * .] (75) 

where L1, L2, L3, etc., are the component lives 

corresponding to 90-percent reliability. 

Example problem. -A gearbox has been designed and 

the component lives have been determined (table 14). 

Find the system life at the 95-percent reliability level. 

Solution.-Iterating the following equation 

lo&$) = (149) 

x [ (&))10/9+ (&y9+ (&J2 

+ (AJ3’b&2+ )2.5+ Go~2~5

TABLE 14.-COMPONENT LIVES FOR 

EXAMPLE PROBLEM 

Gear Weibull Gear 

1 slope, 1 lif”, 
on a programmable calculator yields 

L=249 hr 

for the life at the 95-percent reliability level. This result is 

surprisingly low and emphasizes the need for highly 

reliable components if the required system reliability and 

life are to be met. 

5.4 Helical Gear Life 

The calculation of helical gear teeth life is presented in 

this section. The approach is to think of the helical gear 

as a modified spur gear, with the tooth profile in the 

plane normal to the gear axis being preserved but with the 

spur tooth being sliced like a loaf of bread and wrapped 

around the base circle along a helix of angle ?lr. The 

helical tooth life can be obtained as a “corrected” spur 

tooth life. Adjustment factors for each of the factors that 

influence life in the previous equations are given as 

follows (ref. 88): 

Load adjustment: 

Whelical = 

WspurPb 

0.95 zcos \k (76) 

Curvature sum adjustment: 

c Phelical = (cPspur) COS * 

Face-width-in-contact adjustment: 

Fhelical = Fspur set * 

(77) 

(78) 

No adjustment factor for involute length I is recommended. These adjustment factors are recommended for 

helical gears with axial contact ratios above 2. For lesser 

axial contact ratios the life should be calculated as if the 

gear were a spur gear. 

Example problem.-Calculate the tooth life at 90- 

percent reliability level for a gear with a 3410-N (767-lb) 

force transmitted along the pitch line. The gear has the 

following properties: 

Transverse pressure angle, pt, deg . . . . . . . . . . . . . . . . . . . . . . . . . . 20 

Helix angle, \k, deg.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 

Diametral pitch, P, in-l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 

Module, l/P, mm/tooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,... 3.18 

Number of teeth, N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

Face width, F, mm (in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 (1.000) 

Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard 

Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AISI 93 10 

(The properties in the transverse plane are taken from the 

previous example for spur gear tooth life.) 

Solution.-From the previous example problem, the 

transverse base pitch is 9.38 mm (0.369 in) and the length 

38 

of the contact path is 15.10 mm (0.595 in). The axial pitch 

is calculated as follows: 

p,2.!-= 9.38 mm 

tan \k tan = 9.38 mm (0.369 in) 

Also the axial contact ratio can be evaluated. 

Ftan + Axial contact ratio = ___ 

Pa 

= (25.4mn-O tan (45”) =2 71 

9.38mm . 

Since the axial contact ratio is greater than 2, the 

adjustment factors in equations (76) to (78) should be 

used. 

Whelical = 

(3400 N)(9.38 mm) 

0.95 (15.10mm) cos (45”) = 3.14 kN (706 lb) 

CPhelical =(0.1314 mm-‘) cos (45”) 

=0.0929 mm-r (2.36 in-l) 

Fhelical=(25.4 mm) set (45”)=35.9 mm (1.414 in) 

These adjusted values can now be substituted into the 

tooth life equation (68), to determine the 90-percent life. 

Tu,=(6.44x 109)(3140)-4.3(35.9)3.9 

x (0.0929) 

= 8.85 x 101’ stress cycles 

In hours, the life is 

- 5.0(1.333) -0.4 

Tlo=(8.85 x 10” cycles) 

= 1.48 x 106 hr 

5.5 Bevel and Hypoid Gear Life 

For bevel and hypoid gears the contact pattern can be 

determined from a tooth contact analysis (TCA). The 

TCA is done with a computer program developed by The 

Gleason Works (ref. 95). The mathematics is very 

complicated (refs. 96 and 97), but the needed parameters 
for a life analysis are simply the surface curvatures and 

the path of contact (from which the Hertz stress can be 

calculated). 

Assuming that the result from a TCA analysis is 

available, the bevel or hypoid tooth life is calculated by 

the following formulas, which are adapted from 

reference 98. An alternative method of assessing surface 

durability is presented in reference 99. 

*,,= ( qg)“2.5 (79) 

where Dru is 1.429 x 10j5 for newtons and millimeters or 

3.58 x 1056 for pounds and inches and V= wzf. 

zip=‘+’ +‘+L curvature sum 

Pl,X P2,x Pl,Y P&Y 

(80) 

Ap= (&+ig-C&+i-$) 

&J 

curvature difference (81) 

From table 15, get the values of a*, b*, and ‘I’ 

TABLE 15 .-CONTACT 

STRESS PARAMETERS 

AP a* b’ T 

) 1 

.1075 1.0760 

.3204 1.2623 

.4795 1.4556 

.5916 1.6440 

.9318 

.8114 

.7278 

.6687 

1.2808 

1.2302 

1.1483 

1.0993 

1.0701 

.6716 1.8258 .6245 1.0517 

.7332 2.011 .5881 I .0389 

.7948 2.265 .5480 1.0274 

.83495 2.494 .5186 1.0206 

.87366 2.800 .4863 I .0146 

.90999 3.233 .4499 1.00946 

.93657 3.738 .4166 1.00612 

.95738 4.395 .3830 1.00376 

.97290 5.267 .3490 1.00218 

.983797 6.448 .3150 1.00119 

.990902 8.062 .2814 1.000608 

.995112 10.222 .2497 1.000298 

.997300 2.789 .2232 1.000152 

.9981847 4.839 .2070 1.000097 

.9989156 7.914 .I8822 1.000055 

.9994785 !3.55 .16442 1 .OCQO24 

.9998527 17.38 .1305a 1.000006 

1 03 1 1 .OOOooo 

a=a* (82) 

(2T - l)S, 

7o= 2T(T+ 1) (85) 

b 

z = (T + 1)(2T - 1)“2 

(83) 

(84) 

(86) 

Example problem.-A TCA study showed the contact 

path on the bevel pinion to be approximately 7.6 mm (0.3 

in) in length. At the central point of the contact path the 

principal radii of curvature are as follows (concave 

negative): 

Pinion Gear 

~~,~=50.8 mm (2.00 in) P2,x = O” 

P’,Y = 101.6 mm (4.00 in) p2,y= - 127.0 mm (- 5.00 in) 

The load normal to the tooth surface is 3.11 kN (700 lb). 

Estimate the tooth life for the pinion. 

Solution .-First, using equation (80), calculate the 

curvature sum. 

cp= 1 1 1 1 

50.8 mm 

+ 

m 

+ - 

101.6 mm 127.0 mm 

=0.0217 mm-l (0.550 in-l) 

Next determine the nondimensional curvature difference 

from equation (81). 

Ap= 

1 1 

101.6 mm - 127.0 mm 

0.0217 mm - 1 

= 0.816 

Using the curvature difference and interpolating from 

table 15 yields the following values: 

a* = 2.386 

b* = 0.5325 

I’= 1.024 

39 
Now, determine the dimensions of the Hertz ellipse by 

using equations (82) and (83). Assume the material 

properties of steel: E = 207 GPa (30.0 x 106 psi); v = 0.25. 

)I l/3 

3(3110 N) 1 - 0.252 

)I 

l/3 

2(0.0217mm-l) (2) 207 x 103N/mm2 

= 1.249 mm (0.0492 in) 

a=(2.386)(1.249 mm)=2.98 mm (0.1174 in) 

b = (0.5325) (1.249 mm) = 0.665 mm (0.0262 in) 

Calculate the maximum Hertz compression stress by 

using equation (84). 

S,= 3(3110 N) 

2n(2.98 mm)(0.665 mm) = 749 MPa (108.7 ksi) 

Evaluate the maximum subsurface orthogonal reversing 

shear stress and the depth below the surface at which it 

occurs by using equations (85) and (86), respectively. 

[w .w - 11(749 MW = 182 8 MPa (26 5 ksi) 

r”= 2(1.024)[1.024+1] * 

0.5325 

z= (1.024+ 1)[2(1.024)- ill/2 =oe321 mm (“*0126 in) 

Assume that the semiwidth of the Hertzian contact ellipse 

is representative of the width of the contact path across 

the tooth, and calculate the stressed volume as follows: 

V= wzl-azf = (2.98 mm)(0.321 mm)(7.6 mm) 

= 7.27 mm3 (444 X 10V6 in3) 

Finally calculate the tooth life at go-percent reliability by 

using equation (79). 

(1.429 x 1035)(0.321 mm)7’3 1’2.5 

(182.8 N/mm2)j1/3(7.27 mm3) 1 

= 8080 million stress cycles 

6.0 Lubrication 

In gear applications fluid lubrication serves four 

functions: 

(1) It provides a separating film between rolling and 

sliding contacting surfaces and thus prevents wear. 

(2) It acts as a coolant to maintain proper. gear 

temperature. 

(3) It protects the gear from dirt and other 

contaminants. 

(4) It prevents corrosion of gear surfaces. 

The first two lubrication functions are not necessarily 

separable. The degree of surface separation affects the 

friction mode and the magnitude of friction force. These 

in turn influence frictional heat generation and gear 

temperatures. 

Until the last two decades the role of lubrication 

between surfaces in rolling and sliding contacts was not 

fully appreciated. Metal-to-metal contact was assumed to 

occur in all applications, with attendant required 

boundary lubrication. An excessive quantity of lubricant 

sometimes generated excessive gear tooth temperatures 

and caused thermal failure. The development of the 

elastohydrodynamic lubrication theory showed that 

lubricant films of thicknesses of the order of microinches 

and tens of microinches occur in rolling contacts. Since 

surface finishes are of the same order of magnitude as the 

lubricant film thickness, the significance of gear tooth 

finish to gear performance became apparent. 

6.1 Lubricant Selection 

The useful bulk temperature limits of several classes of 

fluid lubricants in an oxidative environment are given in 

table 16 (ref. 100). Grease lubricants are listed in table 17 

(ref. 100). The heat transfer requirements of gears dictate 

whether a grease lubricant can be used. Grease 

lubrication permits the use of simplified housing and 

seals. 

The most commonly used lubricant is mineral oil, both 

in liquid and grease form (table 18; ref. 100). As a liquid, 

mineral oil usually contains an antiwear or extremepressure (EP) additive, an antifoam agent, and an 

oxidation inhibitor. In grease the antifoam agent is not 

required. 

Synthetic lubricants have been developed to overcome 

some of the harmful effects of lubricant oxidation. 

However, synthetic lubricants should not be selected over 

readily available and invariably less-expensive mineral 

oils if operating conditions do not require them. It is 

usually easier to incorporate synthetic lubricants in a new 

design than to convert an existing machine to their use. 

40 
TABLE 16.-LUBRICANTS 

Type 

Mineral oil 

Diester 

Type II ester 

Triester 

Superrefined and synthetic 

mineral oils 

Fluorocarbona 

Polyphenyl ethera 

=Not recommended for gears. 

Specification Bulk temperature 

limit in air 

MIL-L-6081 

MIL-L-7808 

MIL-L-23699 

MIL-L-9236B 

I 

“C 

93+ 200+ 

149+ 300+ 

218 425 

218 425 

218 425 

288 550 

315 600 

Type Specification 

Aircraft(high speed; 

ball and roller bearing) 

Aircraft (synthetic; 

extreme pressure) 

Aircraft (synthetic; 

molybdenum disulfide) 

Aircraft (general 

purpose; wide 

temperature range) 

Helicopter (oscillating 

bearing) 

Plug value (gasoline 

and oil resistant) 

Aircraft (fuel and oil 

resistant) 

Ball and roller bearing 

(extreme high temperature) 

“F 

6.2 Elastohydrodynamic Film Thickness 

Rolling elements, under which category gear teeth can 

be considered to fall, are separated by a highly 

compressed, extremely thin lubricant film of thickness h. 

Because of the presence of high pressures in the contact 

zone the lubrication process is accompanied by some 

elastic deformation of the contact surface. Accordingly 

this process is referred to as elastohydrodynamic (EHD) 

lubrication. Ertel (ref. 101) and later Grubin (ref. 102) 

were among the first to identify this phenomenon. 

Hamrock and Dowson (ref. 103) have derived a 

simplified approach to calculating the EHD film 

thickness as follows: 

ffmin = 3.63 0.68 Go.49 W-0.073( 1 - e - 0.68k) (87) 

In this equation the most dominant exponent occurs on 

the speed parameter U, and the exponent on the load 

parameter W is very small and negative. The materials 

parameter G also carries a significant exponent, although 

TABLE 17.-GREASES 

MIL-G-38220 

MIL-G-23827 

MIL-G-21164 

MIL-G-81322 

MIL-G-25537 

MIL-G-6032 

MIL-G-27617 -1 to204 -3oto400 

MIL-G-25013 - 73 to 232 - 100 to 450 

- 40 to 400 I I Thickening agent and 

fluorocarbon 

- 100 to 250 Thickening agent, lowtemperature synthetic 

oils, or mixtures of 

EP additive 

-40 to 204 

-73 to 121 

-73 to 121 

-54 to 177 

-54 to 71 

0 to 95 

- 100 to 250 I I Similar to MIL-G-23827 

plus MoS2 

-65 to 350 

-65 to 160 

32 to 200 

Thickening agent and 

synthetic hydrocarbon 

Thickening agent and 

mineral oil 

Thickening agent, 

vegetable oils, glycerols, 

or polyesters 

Thickening agent and 

fluorocarbon and 

fluorosilicone 

Thickening agent and 

silicone fluid 

I I I 

41 
TABLE IS.-MINERAL OIL CLASSIFICATION AND 

COMPARATIVE VISCOSITIES 

Iategory SAE 

lumber 

ASTM 

gradea 

4GMA 

;ear oil 

c Temperature 

38 “C (100 OF) 99 “C (210 OF) 

Approximate viscosity, CS 

32 _--- 2 ---- 

40 ---- 3-5.5 ---_ 

60 _--- 8.5-12 ---- 

75 _--- 12-16 ---- 

105 ---- 19-24 ---- 

150 _--- 29-35 ---- 

E: rctra 

Ii& :ht 

---_ 215 _--- 42-51 ---- 

low 1 45 4.2 

---- 315 _--- 61-75 ---- 

Li ight 2ow ---- 2 69 5.7 

465 _--- 90-I 10 ---- 

M :edium 3ow ---- ---- ------ 10 

---- 3 128 ---- 

---- 700 _--- 130-166 

M ledium 

hc :avy 

40 ---- _--- _----_ 13 

4 183 

H eavy 50 ---- _--- 17 

loo0 _--- 194-237 18 

---- ---- 5 _----- ---- 

1500 _--- 291-356 

2150 _--- 417-525 

---- ---- 6 _----- 24 

140 _--- 25 

3150 _--- 630-780 ---- 

St 

he 

lper ---- “7 29 

:avy ---_ 4650 _--- 910-I 120 

---- ---_ “8 ------ 36 

250 ---- 43 

“8A __---- 47 

---- 7000 _--- 1370-1670 

---_ 9 ------ 97 

---- ---- 10 _----_ 227 

---- ---- 11 _----- 464 

aGrade number is equivalent to average Saybolt Umversal Viscosity (SUS) at 38 ‘C (100 “FL 

bCompounded with fatty oil. 

the range of this variable in engineering situations is 

limited. 

The variables related in equation (87) are from five 

dimensionless groupings: 

Dimensionless materials parameter G = CUE’ (92) 

where 

E’ effective elastic modulus, Pa (psi) 

WN normal applied load, N (lb) 

h film thickness, m (in) 

PX effective radius of curvature in x (motion) 

direction, m (in) 

PY effective radius of curvature in y (transverse) 

direction, m (in) 

Dimensionless film thickness H= h/p, (88) 

Ellipticity parameter k = (pY/px)2/7r 

Dimensionless load parameter W= W$E’pz 

(89) 

(90) 

Dimensionless speed parameter U= ~ou/E’px (91) 

42 
u mean surface velocity in x direction, m/s (in/s) 

a pressure-viscosity coefficient of fluid, m2/N 

(inz/lb) 

I*0 atmospheric viscosity, N s/m2 (lb s/in2) 

and 

2 E’ = (93) 

1 1 1 -=-+- 

PX P1.x P2,x 

1 

-= L+-!- 

PY Pl,Y QY 

(94) 

where 

%% elastic modulus of bodies 1 and 2 

VlyV2 Poisson’s ratio of bodies 1 and 2 

pl,x,p2,x radius in x (motion) direction of bodies 1 and 2 

~~,~,pz,~ radius in y (transverse) direction of bodies 1 

and 2 

For typical steels E’ is 227.5 GPa (33 x 106 psi), and for 

mineral oils a typical value of 01 is 2.2 m2/N (1.5 x 10e4 

ir$/lb). Thus for mineral-oil-lubricated rolling elements, 

G = 5000. 

For bodies in sliding or rolling contact with parallel 

axes of rotation, the tangential surface velocities are 

Ul = V-1,x 

(95) 

v2=~2r2,~ 

where rl,x and r2,x are the radii from the centers of 

rotation to the contact point. 

The geometry of an involute gear contact at distance s 

along the line of action from the pitch point can be 

represented by two cylinders rotating at the angular 

velocity of the respective gears. The distance s is positive 

when measured toward the pinion (member 1). 

Equivalent radius, from equation (94), is 

(rp 1 sin cp - s)(rp 2 sin p + s) 

Px= ’ 

(rp,l + rP,2) sin cp 

(96) 

Contact speeds from equation (87) are 

q = q(rp,l sin cp --s) 

(97) 

02 = P2(rP,2 sin cp + s> 

Surface topography is important to the EHD 

lubrication process. EHD theory is based on the 

assumption of perfectly smooth surfaces, that is, no 

interaction of surface asperities. Actually, of course, this 

is not the case. An EHD film of several millionths of an 

inch can be considered adequate for highly loaded rolling 

elements in a high-temperature environment. However, 

the calculated film might be less than the combined 

surface roughness of the contacting elements. If this 

condition exists, surface asperity contact, surface distress 

(in the form of surface glazing and pitting), and surface 

smearing or deformation can occur. Extended operation 

under these conditions can result in high wear, excessive 

vibration, and seizure of mating components. A surface 

roughness criterion for determining the.extent of asperity 

contact is based on the ratio of film thickness to a 

composite surface roughness. The film parameter A is 

where composite roughness u is 

u= uf+u$ 1’2 

( > 

(99) 

and aI and a2 are the rms roughness of the two surfaces in 

contact. The film parameter A can be used as an indicator 

of relative life (fig. 37). Although the curve has been 

prepared by using the rms estimate, an arithmetic average 

might be used. The A ratio can also be used as an 

indicator of gear performance. At A < 1, surface 

scoring, smearing, or deformation, accompanied by 

wear, will occur on the gear surface. When l< A c 1.5, 

surface distress and superficial surface pitting can occur. 

For 1.5 < A c 3 some surface glazing can occur and 

eventually the gear will fail by classical subsurfaceoriginated rolling-element (pitting) fatigue. At A 1 3 

minimal wear and extremely long life can be expected. 

I I I III1 

.6 .8 1 2 4 6 810 

Film parameter, A 

Figure 37.-Relative life as a function of film parameter. 

43 
Failure will eventually be by classical subsurfaceoriginated rolling-element (pitting) fatigue (ref. 93). 

6.3 Boundary Lubrication 

Extreme-pressure lubricants can significantly increase 

the load-carrying capacity of gears. The extreme-pressure 

additives in the lubricating fluid form a film on the 

surfaces by a chemical reaction, adsorption, or 

chemisorption. These boundary films can be thinner than 

0.025 pm (1 pin) or several microinches thick (ref. 104). 

These films are formed from the chemical reaction of 

sulfur (ref. 105) or from the chemisorption of iron 

stearate. Film thickness varies for various types of film 

(fig. 38, ref. 104). The films can separate the metal 

surfaces when the lubricant becomes thin enough for the 

asperities to interact. The boundary film probably lubricates by microasperity-elastohydrodynamic lubrication 

(ref. 104), where the asperities deform under the load. 

The boundary film prevents contact of the asperities and 

at the same time provides low-shear-strength properties 

that prevent shearing of the metal and reduce the friction 

coefficient below that of the base metal. These boundary 

films provide lubrication at different temperature 

conditions depending on the materials used. For 

example, some boundary films will melt at a lower 

temperature than others (ref. 105) and will then fail to 

protect the surfaces. The “failure temperature” is the 

temperature at which the lubricant film fails. In extremepressure lubrication this failure temperature is the 

temperature at which the boundary film melts. 

[‘Monolayers 

IiExtreme-pressurefilms 

0 Boundary films (?) 

I l=jEHD films 

0 Sliding wear debris 

Rolling wear debris 0 

11 Asperity heights 

Microconjunctions 0 

Macroconjunctions m 

10-S 10-b 10-4 10-Z 

Size, cm 

100 

Figure 38.-Range of film thickness for various types of lubricant film, 

with relation to surface roughness and wear. 

The melting point or thermal stability of surface films 

appears to be a unifying physical property governing 

failure temperature for a wide range of materials (ref. 

105). It is based on the observation (ref. 106) that only a 

solid film can properly interfere with potential asperity 

contacts. For this reason many extreme-pressure 

lubricants contain more than one chemical for protection 

over a wide temperature range. For instance, Borsoff 

(refs. 107 and 108) found that phosphorous compounds 

are superior to chlorine and sulfur at slow speeds, but 

sulfur is superior at high speeds. He explains this as a 

result of the increased surface temperature at the higher 

speeds. (Most extreme-pressure additives are chemically 

reactive and increase their chemical activity as 

temperature is increased.) Horlick (ref. 109) found that 

some metals such as zinc and copper have to be removed 

from lubrication systems when using certain extremepressure additives. 

6.4 Lubricant Additive Selection 

Some of the extreme-pressure additives commonly 

used for gear oils contain one or more compounds of 

chlorine, phosphorus, or sulfur or lead soaps (ref. 110). 

Many chlorine-containing compounds have been suggested as extreme-pressure additives, but few have been 

used. Some lubricants are made with chlorine-containing 

molecules where the Cl,-C linkage is used. For example, 

either tri (trichloroethyl) or tri (trichlortert butyl) 

phosphate additives have shown high load-carrying 

capacity. Other chlorine-containing additives are 

chlorinated paraffin or petroleum waxes and 

hexachlorethene. 

The phosphorus-containing compounds are perhaps 

the most commonly used additives for gear oils. Some 

aircraft lubricants have 3 to 5 percent tricresyl phosphate 

or tributyl phosphite as either an extreme-pressure or 

antiwear agent. Other phosphorus-containing extremepressure agents used in percentages of 0.1 to 2.0 could be 

dodecyl dihydrogen phosphate, diethyl-, dibutyl-, or 

dicresyl-phenyl trichloroethyl phosphite and a phosphate 

ester containing a pentachlorphenyl radical. Most of the 

phosphorous compounds in gear oils also have other 

active elements. 

The sulfur-containing extreme-pressure additives are 

believed to form iron sulfide films that prevent wear at 

high loads and speeds. However, they give higher friction 

coefficients and are therefore usually supplemented by 

other boundary-film-forming ingredients that reduce 

friction. The sulfur compounds should have controlled 

chemical activity (e.g., oils containing dibenzyl disulfide 

of 0.1 or more percent). Other sulfur-containing extremepressure additives are diamyl disulfide, dilauryl disulfide, 

44 
sulfurized oleic acid and sperm oil mixtures, and 

dibutylxanthic acid disulfide. 

Lead soaps have been used in lubricants for many 

years. They resist the wiping and sliding action in gears 

and help prevent corrosion of steel in the presence of 

water. Some of the lead soaps used in lubricants are lead 

oleate, lead fishate, lead-12-hydroxystearate, and lead 

naphthenate. Lead naphthenate is used most often used 

because of its solubility. Lead soaps are used in concentrations of 5 to 30 percent. 

Other additive compounds contain combinations of 

these elements. Most extreme-pressure lubricants contain 

more than one extreme-pressure additive. Needless to 

say, the selection of a proper extreme-pressure additive is 

a complicated process. The word “susceptibility” is 

frequently used with reference to additives in oils to 

indicate the ability of the oil to accept the additive 

without deleterious effects. Such properties as solubility, 

volatility, stability, compatibility, load-carrying capacity, 

and cost must be considered. Many gear oil compounds 

depend on the use of proprietary, or package, extremepressure additives. As a result, the lubricant manufacturer does not evaluate the additives’ effectiveness. 

Because of this, any selection of extreme-pressure 

additives should be supported by an evaluation program 

to determine their effectiveness for a given application. 

However, a few firms have considerable background in 

the manufacture and use of extreme-pressure additives 

for gear lubrication, and their recommendations are 

usually accepted without question by users of gear oils. 

6.5 Jet Lubrication 

For most noncritical gear applications where cooling 

the gear teeth is not an important criterion, splash or mist 

lubrication is more than adequate to provide acceptable 

lubrication. However, in critical applications such as 

helicopter transmission systems and turboprop aircraft 

gearboxes, heat rejection becomes important. Hence jet 

lubrication is used. It is important to have the oil 

penetrate into the dedendum region of the gear teeth in 

order to provide cooling and lubrication. Failure to do so 

can result in premature gear failure, primarily by scoring 

and wear. Methods of oil jet lubrication were analytically 

determined and experimentally verified by Akin and 

Townsend (refs. 111 to 117). 

Out of mesh.-An analysis of out-of-mesh jet 

lubrication of spur gears along with experimental results 

is given in references 111 and 114. The analysis 

determined the impingement depth for both gear and 

pinion when the jet is pointed at the out-of-mesh location 

and directed at the pitch points of the gear and pinion. 

Figure 39 shows the oil jet as it clears the gear tooth and 

Gear 2 Pinion 1 

‘, Hot (load) side of teeth 

(a) 

Jet 

(b) 

Jet 

(a) Gear tooth at position where oil jet is just clearing gear tooth and 

begins flight toward pinion tooth. Pinion shown in position a. 

(b) Moving jet from point c on gear to point d on pinion, the rotates 

pinion from position a to b. 

Figure 39.-Model for out-of-mesh jet lubrication. 

impinges on the pinion tooth and also gives the 

nomenclature for the equations. The time of flight for the 

oil jet to clear the gear tooth and impinge on the pinion 

tooth must equal the time of rotation of the gears and is 

given by the following equation: 

Time of rotation = Time of flight 

wi’ tan-12 

[ 

’ 

rP,l 

+lnv (gg$ 

( > 

-% 

x 

( 

cos-1: -inv (ocZ+inv (a -inv (p 

a, > 1 

= (<,2 - &Y2-x1 (100) 

Uj 

where 

inv (o= tan p-cp 

This equation must be solved iteratively for x1 and 

substituted into the following equation to determine the 

pinion impingement depth. 

dl = r,,l- ( > +x-j 1’2 

45 
A similar equation is used to determine the time of flight 

and the time of rotation of the gears for impingement on 

the unloaded side of the gear tooth. 

WF1 (y +inv ~cos-1[‘o,1~~,2+‘l~1’2]j 

_ cos-‘(rP,yT,,r)-invcp,,t +inv (0 -inv 

mg 

cp 

> 

= (I& -$Jy2- 2 

J 

Solving for x2 interatively and substituting 

following equation gives the impingement depth 

gear: 

(102) 

in the 

for the 

(103) 

At high gear ratios it is possible for the gear to shield the 

pinion entirely so that the pinion would receive no 

cooling unless the jet direction or position were modified. 

The following three tests can be used to determine the 

condition for zero impingement depth on the pinion, 

when 

tan-12 < cos-1-!2- 

rl r&l 

or 

x1 < @,I -,1y2 

and when dl I 0. 

The pinion tooth profile is not impinged on directly for 

speed reducers with ratios above 1.11 to 1.23, depending 

on the jet velocity ratio. Gear impingement depth is also 

limited by leading tooth blocking at ratios above 1.25 to 

1.5. 

Into me&-For into-mesh lubrication there is an 

optimum oil jet velocity to obtain best impingement 

depth and therefore best lubrication and cooling for the 

gear and pinion (refs. 116 and 117). This optimum oil jet 

velocity is equal to the gear and pinion pitch-line velocity. 

When the oil jet velocity is greater or less than the pitchline velocity, the impingement depth will be less than 

optimum. Also the oil jet should be pointed at the 

intersections of the gear and pinion outside diameters and 

should intersect the pitch diameters for best results. A 

complete analytical treatment for into-mesh lubrication is 

given in references 117 and 118. Into-mesh oil jet 

lubrication gives much better impingement depth than 

out-of-mesh jet lubrication. However, in many applications, considerable power is lost from oil being trapped in 

46 

the gear mesh. Radial oiI jet lubrication at the out-ofmesh location gives the best efficiency, lubrication, and 

cooling (ref. 112). 

Radial oil jet lubrication.-The vectorial model from 

reference 114 for impingement depth when using a 

radially directed oil jet is shown in figure 40. The 

impingement depth d can be calculated to be 

d= 1.57+2tan (0+8/2 

P[nD/(2977*) + tan cp] 

(104) 

where the oil jet velocity uj= 13 r@ (m/s; ft/s) and Ap is 

the jet pressure in pascals (psig). If the required d is 

known, the Ap that is required to obtain that 

impingement depth is 

ndPD 

3 

2 

Ap= 2977[1.57 + B/2 + (2 - dP) tan cp] (105) 

High-speed motion pictures were used to determine 

experimentally the impingement depth for various oil 

pressures and gear rotational speeds. Figure 41 is a plot 

of calculated and experimental impingement depth versus 

jet nozzle pressure for gear speeds of 2560 and 4920 rpm. 

At both speeds there is good agreement between the 

calculated and experimental impingement depths at 

higher pressures. However, at lower pressures, there is 

considerable difference between the calculated and 

experimental impingement depths. Most of this 

difference in impingement depth is due to viscous losses 

in the nozzle with the very viscous oil used. 

Oil jet -,,,r*, 

\ \ 

‘, (1.57+2tan(o+BlZ) 

P 

Figure 40.-Vectorial model for penetration depth. 
Temperature, 

K 

Gear rotational 

speed, 

rpm 

Oil jet pressure, Pa 

0 - 60 

Oil jet pressure, psi 

Figure 41 .-Calculated and experimental impingement depth as a 

function of oil jet pressure at gear rotational speeds of 4920 and 2560 

rpm 

6.6 Gear Tooth Temperature 

A computer program to predict gear tooth temperatures was developed by using finite-element analysis (refs. 

83, 84, and 118). However, this program does not include 

the effects of oil jet cooling and oil jet impingement 

depth. The program used an average surface heat transfer 

coefficient for surface temperature calculation that was 

based on the best information available at that time. 

A method for predicting gear tooth temperature more 

accurately must be based on an analysis that allows for 

the use of a heat transfer coefficient for oil jet cooling 

coupled with a coefficient for air/oil mist cooling for that 

part of the time that each condition exists. Once the 

analysis can make use of these different coefficients, it 

can be combined with a method that determines the oil jet 

impingement depth to give a more complete gear 

temperature analysis program (ref. 115). However, both 

the oil jet and air/oil mist heat transfer coefficients are 

unknowns and must be determined experimentally. Once 

the heat generation and the oil jet impingement depth 

have been calculated, the heat transfer coefficients are 

either calculated or estimated. Then a finite-element 

analysis can be used to calculate the temperature profile 

of the gear teeth (fig. 42). The differential profiles in this 

figure are the temperature differences between the 

calculated gear tooth temperatures and the inlet cooling 

oil temperature. Infrared temperature measurements of 

the gear tooth during operation verified the accuracy of 

the calculated values for the gear tooth surface (refs. 112 

and 115). 

7.0 Power Loss Predictions 

An experimental plot of efficiency as a function of 

input torque for a 236-kW (317-hp), 17-to-1 ratio helicopter transmission system comprising a spiral-bevel gear 

(a) 

11 

tb) 

(a) Zero impingement depth. 

(b) 87.SPercent impingement depth. 

Figure 42.-Calculated gear tooth temperature increase above oil inlet 

temperature. Gear rotational speed, 10 00 rpm; load, 5903 N/cm 

(3373 lb/in); pitch diameter gear, 8.9 cm (3.5 in.) 

input and a three-planet system is shown in figure 43. The 

nominal maximum efficiency of this transmission is 98.4 

percent. The power loss, 1.5 percent of 235 kW (317 hp), 

is highly dependent on load and almost independent of 

speed. Table 1 gives nominal efficiency values for various 

types of gear system. By multiplying the efficiency for 

each gear stage, the overall nominal efficiency of the gear 

system can be estimated. Unfortunately most 

transmission systems do not operate at maximum rated 

torque. As a result, the percentage of power loss at part 

load will generally be high. In addition, gear geometry 

can significantly influence power losses and, in turn, the 

efficiency of gearsets. 

As an example, consider a set of spur gears with a 

lo-cm (4-in) pinion. A reduction in gear diametral pitch 

from 32 to 4 can degrade peak efficiency from 99.8 to 

99.4 percent under certain operating conditions. 

Although this reduced efficiency may at first glance 

appear to be of little significance, the change does 

represent a 200-percent increase in power loss. 

Although some analytical methods have been 

developed for predicting gear power losses, most of them 

do not conveniently account for gear mesh geometry or 

operating conditions. Generally these predictive 

47 
98. 2 

c 

Gear rotational 

speed, 

rpm 

0 6060 

200 300 400 

Input torque, N-m 

I 

1.5 2.0 2.5 3.0 3.5x103 

Input torque, in-lb 

Figure 43.-Experimental efficiency as a function of input torque for 

three-planet (236-kW; 317-hp) helicopter transmission with spiralbevel gear input and input-output ratio of 17. 

techniques base power loss estimates on the friction 

coefficient at the mesh, among other parameters. 

Experience shows that the accuracy of these analytical 

tools often leaves much to be desired, especially for 

estimating part-load efficiency. 

An evaluation of all of the power loss components as a 

function of torque and speed (fig. 44) shows that an 

unloaded gearset rotating at moderate to high speeds can 

account for more than half of the total power losses at 

full load. Although sliding loss is dominant at low 

operating speeds, it becomes only a moderate portion of 

the total gear system losses at higher speeds. Both rolling 

loss and support ball bearing loss increase with operating 

speed. Good estimates of these speed-dependent losses 

are vital for accurately determining the power consumption of gearboxes. Anderson and Loewenthal (refs. 119 

to 122) have developed an analytical method to 

accurately predict power losses in spur gearsets that has 

been correlated with experimental data. Their method 

applies to spur gears that have standard tooth proportions and are jet or splash lubricated. Moreover, it 

considers the individual contributions of sliding, rolling, 

and windage to power loss at the gear mesh. Some of the 

power loss computations required by the method involve 

the use of mathematical expressions based on average 

operating conditions where the friction coefficient f, 

average sliding velocity v,, average total rolling velocity 

Ur, and EHD film thickness h are all evaluated at the 

point halfway between the pitch point and the start of 

engagement along the line of action. The mesh losses are 

determined simply by summing the siding, rolling, and 

windage power loss components. Thus 

0 Sliding 

0 Roiling m Gear windage 

m Pinion bearing m Pinion windage 

&@ Gear bearing Pinion and gear windage 

Pinion torque, N-m 

Pinion torque, ft-lb 

(a) Pinion speed, 250 rpm. 

(b) Pinion speed, 1000 rpm. 

(c) Pinion speed, 2000 rpm. 

Figure 44.-Typical power loss breakdown for a gear system. Gearset 

parameters include pitch diameter, 15.2 cm; gear ratio, 1.66; 

diametral pitch, 8; ratio of pinion face width to diameter, 0.5; 

lubricant viscosity, 30 cP. 

Q,=Qs+Qr+Qw,l+Qw,z (106) 

Note that the churning loss of gears running submerged 

in oil is not accounted for in this analysis. 

7.1 Sliding Loss 

The sliding loss Q, at the mesh results from frictional 

forces that develop as the teeth slide across one another. 

This loss can be estimated as 

where values of the constants Cl to CT are given in table 

19 and friction coefficient f, average normal load %,v, 

and average sliding velocity is are 

f =0.0127 log 

(7 

c,w, 

Fw& 

(108) 

(1W 

48 
TABLE 19.-CONSTANTS FOR COMPUTING 

GEAR MESH POWER LOSSES 

Value for 

$1 (metric) units 

2~ 10-3 

9x104 

1.16x 1O-8 

0.019 

39.37 

29.66 

2.05 x 10-7 

Value for 

U.S. customary units 

3.03 x 10-4 

1.970 

1.67x lo-l4 

2.86x 1O-9 

1.0 

45.94 

4.34 x IO-3 

v,= I&-5 (in the vector sense) 

2, m/s US/s) (110) 

Note that the expression for f is based on test data 

applied to gear sliding-loss computations involving the 

elastohydrodynamic (EHD) lubrication regime, where 

some asperity contact occurs. Such a lubrication regime 

generally can be considered as the common case for 

gearset operation. Parameter f should be limited to the 

range 0.01 to 0.02. For computations of is and f, the 

length of the line of action Z and the average total rolling 

velocity ir are 

z=o.5 [[(D1+ &)2-(D1 COS d2-j l/2 

+ [ (D2+ &-)2-tD2cos d2] 1’2 

-(Dl +DZ, sin cp (111) 

7.2 Rolling Loss 

Rolling loss occurs during the formation of an EHD 

film when oil is squeezed between the gear teeth and then 

pressurized. The rolling loss Q, is a function of the EHD 

film thickness h and the contact ratio m,, which denotes 

the average number of teeth in contact. Therefore 

Q, = C2hv,Fm, (113) 

where the “central” EHD film thickness h for a typical 

mineral oil lubricating steel gears is 

h = C7(~~p)0.67w~0.067p&$4 (114) 

and the contact ratio is 

CSZP 

mc= ~ 7rcoscp (119 

The expression for h in equation (114) does not consider 

the thermal and starvation effects occurring at high 

operating speeds, generally above a pitch-line velocity of 

40 m/s (8000 ft/min). These effects typically tend to 

constrain the film thickness to some limiting value. For 

computing h the equivalent-contact rolling radius is 

Dl(sin cp) + G I I Dz(sin cp) - --$ 

Peq = (116) 

2(D1 +Dd sin 9 

7.3 Windage Loss 

In addition to sliding and rolling losses, power losses 

also occur from pinion and gear windage. Such losses can 

be estimated from expressions based on turbine-disk drag 

test data. Thus the pinion and gear windage losses are 

approximated as 

&=C,(l+4.6$) ($2’8 

x H2.6(0.028 p + CA)O.~ (118) 

These windage loss expressions apply for the air/oil 

atmosphere commonly found in gearboxes. The 

expressions assume an oil specific gravity of 0.9 and 

constant values for air density and viscosity at 66 “C 

(150 “F). To further account for the oil-rich gearbox 

atmosphere, both the density and viscosity of the 

atmosphere are corrected to reflect a 34.25to-1 air/oil 

combination, which often is reported for helicopter 

transmissions lubricated with oil jets. 

49 
7.4 Other Losses 

To determine the total gear system loss, power losses 

due to the support bearings should also be considered. 

Both hydrodynamic and rolling-element bearing losses 

often can equal or exceed the gear mesh losses. Bearing 

losses can be quantified either theoretically or by testing. 

Total gear system loss is 

QT=Q~+QB (119) 

where 

Q, gear mesh losses 

QB total bearing power losses 

Given QTand input power Qj,, gear system efficiency in 

percent is predicted to be 

(= Q'"xQr,oo 

Qin 

y;dc&n 

diameter, 

(120) 

7.5 Optimizing Efficiency 

The method described for predicting spur gear power 

losses can also be employed in a repetitive manner for a 

parametric study of the various geometric and operating 

variables for gears. Typical results of such parametric 

studies are summarized by the two efficiency maps 

commonly called carpet plots, shown in figure 45, for 

light and moderate gear tooth loads. These carpet plots 

describe the simultaneous effects on gear mesh efficiency 

when the diametral pitch, pinion pitch diameter, and 

pitch-line velocity are varied. The computed gear mesh 

efficiencies do not include support-bearing losses. 

In these carpet plots the three key variables are 

represented along orthogonal intersecting planes for 

three values of diametral pitch, pinion pitch diameter, 

and pitch-line velocity. The gear mesh efficiency 

accompanying any combination of these three gear 

parameters is represented by an intersecting point for the 

three planes. Efficiencies at intermediate values can be 

readily found by interpolating among the planes. 

Pitchline 

velocity, 

m/s fftlmin) 

cm (in) 6.3 (16b-7 ,- 1.3 (2001 

98 

96 

E 

E 92 1 (a) h 

d 

.E 100.0 

2 

ix 

99.6 

99.2 - 

98.8 

: 98.4 (b) 

(8120) 

,r- 20.3 (4060) r 5.1 (1020) 

(a) Light load. K=70 kPa (10 psi). 

(b) Moderate load. K=2100 kPa (300 psi). 

Figure 45.-Parametric study for light and moderate gear loads. Gearset parameters include gear ratio, 1; ratio of pinion face width to diameter 

0.5; lubricant viscosity, 30 cP. 

50 
The light and moderate spur gear loads are specified in 

terms of a load intensity factor, usually called a K factor. 

The K factor is a widely used parameter that normalizes 

the degree of loading on gears of different size and ratio. 

Essentially the K factor describes the load intensity on a 

gear tooth. It is evaluated as 

(121) 

Two carpet plots (fig. 45) were generated for light and 

moderate loads where Kequals 69 and 2100 MPa (10 and 

300 psi), respectively. Allowable K factors for spur gears 

generally range from about 700 MPa (100 psi) for lowhardness steel gears with generated teeth to 7000 MPa 

(1000 psi) for aircraft-quality, high-speed gears that are 

case hardened and ground. A nominal K factor for a 

general-purpose industrial drive with steel gears with a 

Brine11 hardness of 300, carrying a uniform load at a 

pitch-line velocity of 15 m/s (3000 ft/min) or less, 

typically ranges from 1900 to 2200 MPa (275 to 325 psi). 

For light loads, increasing diametral pitch (resulting in 

smaller gear teeth) and decreasing pitch-line velocity tend 

to improve gear mesh efficiency. For moderate to heavy 

loads, increasing both diametral pitch and pitch-line 

velocity improves efficiency. The reason for this reversal 

in efficiency characteristics is that speed-dependent losses 

dominate at light loads but sliding loss is more 

pronounced at higher loads. Also, at higher speeds, the 

sliding loss is reduced because the sliding friction 

coefficient is lower. Essentially these two parametric 

studies indicate that large diameter, fine-pitch gears are 

most efficient when operated under appreciable load and 

high pitch-line velocity. Consequently, to attain best 

efficiency under these operating conditions, the gear 

geometry should be selected as noted. 

8.0 Optimal Design of Spur Gear Mesh 

When designing gear systems, it is cost effective to 

make the design efficient and long lived with a minimum 

weight-power ratio. This introduces the concept of optimization in the design, that is, selecting as a point of 

reference either minimum size (which relates also to 

minimum weight) or maximum strength. Although the 

designer is seeking to optimize some criteria index, 

he/she must also meet all of the basic design requirements (constraints) such as speed ratio, reliability, 

producibility, acceptable cost, and no failures due to 

wear, pitting, tooth breakage, or scoring. The design 

process must carefully assess all of the possible modes of 

failure and make adjustments in the allowable design 

parameters in order to achieve the optimum design. 

Figure 46 presents a qualitative diagram of expected 

Fatigue breakage 

Speed 

Figure 46.-Various failure regimes encountered by gear teeth. 

failure modes for operation in various regimes of load 

and speed. The job of the designer is to anticipate the 

type of load and speed conditions the gears will see in 

service and to choose a design that will not fail and is 

optimum in some sense. 

8.1 Minimum size 

The index for optimization is often minimum size. A 

measure of size is center distance. A procedure for 

optimization based on minimum center distance has been 

developed (refs. 123 and 124). In the procedure the 

following items were the constraints: (1) an allowed 

maximum Hertz (normal) stress S,, of 1.4 GPa (200 000 

psi), which affects scoring and pitting; (2) an allowed 

bending stess Sb of 0.4 GPa (60 000 psi), which affects 

tooth breakage due to fatigue; (3) a geometric constraint 

for no involute interference, which is contact between the 

teeth under the base circle or tip fouling in the case of 

internal gears; and (4) a ratio of tooth width to pinion 

pitch diameter of 0.25, which helps to keep the 

misalignment factor to a minimum. The results of the 

optimization study for a gear ratio of 1 are shown in 

figure 47. After all of the design parameters were 

considered, it was found that, for a given set of allowed 

stresses, gear ratios, and pressure angles, the design space 

could be shown on a plane, with the vertical axis being 

the number of teeth on the pinion and the horizontal axis 

being the diametral pitch. Figure 47 shows the region for 

designs that meet acceptable conditions, but not all of the 

designs resulting from those combinations of number of 

teeth and pitch approach a minimum-center-distance 

design. 

The combinations of number of teeth and diametral 

pitch that fall on any straight line emanating from the 

origin will give a constant-size design. The slope of 

the line is directly related to the center distance. The 

minimum-slope line that falls into the region of acceptable designs gives the optimum choice of pitch and tooth 

number for an optimally small (light) design. 

The data in figure 48 summarize results from a large 

number of plots similar to figure 47. These data can be 

used to select trial values of designs that can lead to 

optimum final designs. The value of these charts is that 

51 
T C = constant 

50 

- Pitting x 

- -- Scoring 

--- Bending 

- //gfy‘ / I Acceptable / Pv 

\\\\\ \ \ \ \ \ 

involute interference 

1 

10 20 30 

Diametral pitch, P 

Figure 47.-External gear mesh design space. Ratio of face width to 

diameter of pitch circle, 0.25; pinion torque, 113 N-m (1000 in-lb); 

maximum Hertz stress, 1.38 GPa (200 x lo3 psi); bending stress, 414 

MPa (60x lo3 psi); Young modulus, 205 GPa (30 x 106 psi), 

Poisson’s ratio, 0.25. 

- 

Pressure 

- ‘i_ , lnti’: 

r 

External 

2 4 6 8 10 

Gear ratio 

(a) Pressure angle, 20. 

(b) Pressure angle, 25. 

Figure 48.-Optimal number of pinion teeth. 

initial designs may be quickly selected. A more detailed 

study of the design should then be made, using the more 

detailed methods for calculating bending fatigue stress, 

life, scoring, and efficiency previously described. Point A 

in figure 47 is the point nearest optimum with respect to 

minimum-center-distance criteria since that is the point 

that lies on the minimum-slope straight line through the 

origin and yet contains points in the regions of acceptable 

designs. Point B is the point that would be selected on the 

basis of only the bending strength criterion and kinematic 

conditions of no involute interference. 

Hertz stress is common to both gear and pinion teeth, 

but the bending strength criterion was applied only to the 

pinion since it is the weaker element of the pinion and 

gear teeth. 

Other approaches to optimization have included the 

balanced-strength approach, where the pinion was made 

thicker and with a shorter addendum whereas the gear 

was made with a longer addendum. The idea was to 

balance the strength of the pinion and gear teeth. In 

reference 125 this idea of balanced design was applied to 

scoring temperature index balance as well as to bending 

strength balance. In reference 126 long- and shortaddenda gearsets were examined for their optimum tooth 

numbers. In reference 127 a direct-search computer 

algorithm was applied to the minimum-center-distance 

design problem. 

8.2 Specific Torque Capacity 

A criterion for comparing two designs has been 

described in the literature (ref. 21) as specific torque 

capacity (STC). The STC rating of a single gear is defined 

as the torque transmitted by that gear divided by the 

superficial volume of its pitch cylinder. That is, torque 

per unit volume is equal to 

T 4T -ZZ 

V rD2F 

where 

(122) 

D pitch circle diameter 

F face width 

For a pair of gears with a ratio of 1 the same STC value 

attributed to each gear individually can be attributed to 

the pair. Thus for such a gear pair the average STC value 

is 

T,/V, + T2/V2 = (4 T, /?rDfFl + 4 T2/rD$Fz) 

2 2 (123) 

52 
But if Tr and Tp are the torque loadings of a pinion and a 

wheel, T2=mGTI, where mG is the gear ratio. Similarly 

D2=m&, but F2=F1. Thus 

or (124) 

SK,= (&) (%) 

But T= WtR/2, where W, is the total tangential load 

acting at the pitch surface. Thus 

sTG(A) ”= ($g (125) 

This expression bears a close resemblance to the AGMA 

K factor for contact stress, where 

K=k%)(~) (126) 

Hence 

STC,, = + 

Thus, although K has been considered as a surface 

loading criterion for a gear pair, it also has a direct 

relationship to the basic torque capacity of a given 

volume of gears. It can therefore be used as a general 

comparator of the relative load capacity of gear pairs, 

without any specific association with tooth surface or 

root stress conditions. The STC value thus becomes a 

useful quantity for comparing the performance of gears 

that are based on completely different tooth systems, 

even where the factors that limit performance are not 

consistent. 

9.0 Gear Transmission Concepts 

9.1 Series Trains 

Speed reducers.-The overall ratio of any reduction 

geartrain is the input shaft speed divided by the output 

TABLE 20.-SPEED RATIOS WITH 

MULTIPLE MESHES 

Type of gear 

Helical and double-helical, 

herringbone gears: 

Double reduction 

Triple reduction 

Ratios 

10-75 

75-350 

Spiral bevel (first reduction) 

with helical gear: 

Double reduction 

Triple reduction 

9-50 

50-350 

Worm gears 

(double reduction) 

100-8000 

Helicals with worm gears 

(double reduction) 

50-270 

Planetary gearsets: 

Simple 

Compound 

4-10 

lo-125 

speed. It is also the product of the individual ratios at 

each mesh, except in planetary arrangements. The ratio is 

most easily determined by dividing the product of the 

numbers of teeth of all driven gears by the product of the 

numbers of teeth of all driving gears. By manipulating 

numbers any desired ratio can be obtained, either exactly 

or with an extremely close approximation. In multiplemesh series trains the forces transmitted through the gear 

teeth are higher at the low-speed end of the train. 

Therefore the pitches and face widths of the gears are 

usually not the same throughout the train. In instrument 

gears, which transmit negligible power, this variation 

may not be necessary (ref. 14). Table 20 gives several 

possible speed ratios with multiple meshes. 

Speed increasers.-Speed-increasing geartrains require 

greater care in design, especially at high ratios. Because 

most gear-sets and geartrains are intended for speed 

reduction, standards and published data in general apply 

to such drives. It is not safe to assume that these data can 

be applied without modification to a speed-increasing 

drive. Efficiency is sometimes lower in an increasing 

drive and substantial input torque is required to overcome output load; in extreme instances self-locking may 

occur (ref. 14). Traction and hybrid drives should be 

considered as reasonable alternatives to gear drives for 

this purpose. 

Reverted trains.-When two sets of parallel-shaft gears 

are so arranged that the output shaft is concentric with 

the input, the drive is called a reverted train. The 

requirement of equal center distance for the two trains 

complicates determination of how many teeth should be 
in each gear to satisfy ratio requirements with standard 

pitches. Helical gears provide greater design freedom 

through possible variation of helix angle (ref. 14). 

9.2 Multispeed Gears 

Sliding gears.-Speed is adjusted by sliding gears on 

one or more intermediate parallel shafts (fig. 49). 

Shifting is generally accomplished by disengaging the 

input shaft. Sliding-gear transmissions are usually 

manually shifted by means of a lever or a handwheel. A 

variety of shaft arrangements and mountings are 

available (ref. 128). 

Constant me&.-Several gears of different sizes are 

mounted rigidly to one shaft mesh with mating gears free 

to rotate on the other shaft (fig. 50). Speed is adjusted by 

locking different gears individually on the second shaft 

by means of splined clutches or sliding couplings. 

Constant-mesh gears are used in numerous applications, among them heavy-duty industrial transmissions. This arrangement can use virtually any type of 

gearing (e.g., spur, helical, herringbone, and bevel gears). 

Manually shifted automotive transmissions combine two 

arrangements. The forward speeds are constant-mesh 

helical gears; the reverse speeds use sliding spur gears 

(ref. 129). 

Idler gears.-An adjustable idler-gear speed drive 

consists of one shaft that carries several different-size, 

rigidly mounted gears (fig. 51). Speed is adjusted through 

an adjustable arm, which carries the idler gear, 

connecting with fixed or sliding gears on the other shaft. 

Sleeve connerted to ~Fixed output gears I \ 

main poYer lource-, 

Input 

shaft 

Figure 49.-Sliding gears. 

Clutch 3,. 

_ outp!li 

shaft 

-- 

--AL Fixed gears 

Figure 50.-Constant-mesh gears. 

output 

shaft 

Input 

shaft e 

!I 

Movab 

idler - 

e 

gear 

Figure 51 .-Idler gear, 

This arrangement is used to provide stepped speeds in 

small increments and is frequently found in machine 

tools. A transmission of this type can be connected to 

another multispeed train to provide an extremely wide 

range of speeds with constant-speed input. Shifting is 

usually manual. The transmission is disengaged and 

allowed to come to a stop before the sliding idler gear is 

moved to the desired setting (ref. 129). 

9.3 Epicyclic Gears 

An epicyclic geartrain (fig. 52) is a reverted-gear 

arrangement in which one or more of the gears (planets) 

moves around the circumference of coaxial gears that 

may be fixed or rotating with respect to their own axes. 

The planet gears have a motion consisting of rotation 

about their own axes and rotation about the axes of the 

coaxial gears (ref. 14). The arrangement shown in figure 

52 consists of a central sun gear with external teeth, a ring 

gear with internal teeth, revolving planet pinions that 

engage the sun gear and the internal ring gear, and a 

planet carrier ‘in which the planet pinions are supported 

(ref. 14). Epicyclic trains may incorporate external or 

internal spur or helical gears or bevel gears arranged in 

numerous ways for fixed-ratio or multispeed applicaPlanet carrier - 

Figure 52.-Planetary gears. Epicyclic gears. 

Sun gear 

Ring gear 

54 
tions. A fixed-ratio, single epicyclic 373-kW (500-hp) 

helicopter transmission using a four-planet arrangement 

with a bevel gear input stage is shown in figure 53. 

Epicyclic trains.-For a single epicyclic train (fig. 

54(a)) the coaxial sun and ring gears are connected by a 

single intermediate planet gear carried by a planet carrier 

rPlanet (NpJ 

r Planet 1NP) \ 

Sun ! 

(Ns) I 

Sun-, \ 1 \ 

arm. For a compound epicyclic train (fig. 54(b)) the 

@I+@ 

+ - - 

intermediate planet pinions are compound gears. Table 

20 gives the speed ratios that can be attained. 

For single planetary arrangements, to make assembly 

possible, (NR +Ns)/q must be a whole number, where NR 

and Ns are the numbers of teeth in the ring and sun gears 

and q is the number of planet gears equally spaced 

around the sun gear (ref. 14). To make assembly possible 

for a compound planetary arrangement (NRNP,,- 

N$Vp,R)/q must equal a whole number (ref. 14), where 

‘-Ring(NR! ‘-Ring (NR) 

(a) (bj 

(a) Simple train. 

(b) Compound train, planet gears keyed to same shaft. 

Figure 54.-Epicyclic gear trains. 

354 rpm 

/ rDummy mast 

, 

near - I \ ,r Mast tandem ball bearing 

1’ / 

30.Scm 

(12.0 in) 

Spiral-bevel gear j 

, - ,“,cl,L LYllCl 

\ bearing 

/ 

,-Planetary support bearing 

- 61 180 rpm 

- \ \ - ‘L 

‘-Input pinion 

Spiral-bevel 

rnlkar howinn 

pinion 

iGearshaft duplex bearing 

Figure 53.-373-kW (NO-hp) helicopter transmission comprising a spiral-bevel gearset and four planet systems. 

55 
NP,s and N~,R are the numbers of teeth in the planet 

gears in contact with the sun and ring gears, respectively. 

Coupled epicyclic trains.-Coupled epicyclic trains 

consist of two or more single epicyclic trains arranged so 

that two members in one train are common to the 

adjacent train (ref. 14). 

Multispeed epicyclic trains.-Multispeed epicyclic 

trains are the most versatile and compact gear 

arrangement for a given ratio range and torque capacity. 

Tables 20 and 21 show the speed ratios possible with 

simple and compound epicyclic trains when one member 

is fixed and another is driving. With a suitable 

arrangement of clutches and brakes an epicyclic train can 

be the basis of a change-speed transmission. With the 

gears locked to each other an epicylic train has a speed 

ratio of 1 (ref. 14). 

Most planetary gear transmissions are of the 

“automatic” type, in which speed changes are carried out 

automatically at a selected speed or torque level. Such 

transmissions are expensive transmissions because of the 

TABLE 21.-SIMPLE EPICYCLIC 

TRAIN RATIOSa 

[See fig. 54(a).] 

Condition Revolution ofsun Arm Ring 

Sun fixed 0 1 I + Ns/NR 

Arm fixed 1 0 - Ns/NR 

Ring fixed 1 +NR/Ns 1 0 

aNs and .VR denote either pitch diameter or number of 

teeth in respective gears. 

clutching and braking elements necessary to control 

operation of the unit. In addition, practical ratios 

available with planetary sets are limited (ref. 128). 

Bearingless planetary.-The self-aligning bearingless 

planetary transmission (fig. 55) covers a variety of 

planetary gear configurations that share the common 

characteristic that the planet carrier, or spider, is 

eliminated, as are conventional planet-mounted bearings. 

The bearings are eliminated by load balancing the gears, 

which are separated in the axial direction. All forces and 

reactions are transmitted through the gear meshes and 

contained by simple rolling rings. The concept was first 

demonstrated by Curtis Wright Corp. under sponsorship 

of the US. Army Aviation Research and Development 

Command (refs. 129 and 130). 

Hybrid transmissions.-Geared planetary transmissions have a limitation on the speed ratio that can be 

obtained in a single stage without the planets interfering 

with each other. For example, a four-planet drive would 

have a maximum speed ratio of 6.8 before the planets 

interfered with each other. A five-planet drive would be 

limited to a ratio of 4.8, and so on. 

A remedy to the speed ratio and planet number 

limitations of simple, single-row planetary systems was 

devised by A.L. Nasvytis (ref. 131). His drive system used 

the sun and ring roller of the simple planetary traction 

drive but replaced the single row of equal-diameter planet 

rollers with two or more rows of stepped, or dualdiameter, planets. With this new, multiroller 

arrangement practical speed ratios of 250 could be 

obtained in a single stage with three planet rows. 

Furthermore the number of planets carrying the load in 

parallel could be greatly increased for a given ratio. This 

.I 

TOutput ring gear 

Inner 

lnout sun .-y’TT 

gear -----kkG$,,ing 

Fixed ring 

(a) ring (bl 

(a) Radial plane. 

(b) Transverse plane. 

Figure 55.-373-kW (500-hp) bearingless planetary transmission. 

56 
resulted in a significant reduction in individual roller 

contact loading with a corresponding improvement in 

torque capacity and fatigue life. 

To further reduce the size and the weight of the drive 

for helicopter transmission applications, pinion gears in 

contact with a ring gear were incorporated with the 

second row of rollers (fig. 56). The ring gear was 

connected through a spider to the output rotor shaft. The 

number of planet roller rows and the relative diameter 

ratios at each contact are variables to be optimized 

according to the overall speed ratio and the uniformity of 

contact forces. The traction-gear combination is referred 

to as the “hybrid transmission.” The potential advantages of the hybrid transmission over a conventional 

planetary transmission are higher speed ratios in a single 

stage, higher power-weight ratios, lower noise (by 

replacing gear contacts with traction rollers), and longer 

life because of load sharing by multiple power paths. 

9.4 Split-Torque Transmissions 

The power-weight ratio of a transmission and the unit 

stress of the gear teeth can be decreased by load sharing 

through multiple power paths. This concept, referred to 

as the “split-torque transmission” (refs. 132 to 134), is 

illustrated in figure 57 for single-input and double-input 

variants. Instead of a planetary gear arrangement the 

input power is split into two or more power paths and 

recombined in a bull gear to the output power (rotor) 

shaft. This concept appears to offer weight advantages 

over conventional planetary concepts without highcontact-ratio gearing. Incorporating high-contact-ratio 

gearing into the split-torque concept is expected to 

further reduce transmission weight. 

9.5 Differential Gearing 

Free differential gearing is an arrangement in which the 

normal ratio of the unit can be changed by driving into 

the unit with a second drive. This arrangement, or one 

having two outputs and one input, is used to vary the 

speed ratio. Simple differentials may use bevel gears (fig. 

58(a)) or spur gears (fig. 58(b)). The bevel gear is used in 

automotive rear-end drives. The input-output speed 

relationship is 

2w=wr-02 

where 

(127) 

0 speed of arm 

ol,w2 shaft speeds 

(a) tb) 

(a) Single input. 

(b) Dual input. 

Figure 57.-Conceptual sketch of split-torque transmission. 

Figure 56.-373-kW (SOO-hp) hybrid transmission. 

57 
TABLE 22.-COMPOUND EPICYCLIC TRAIN RATIO9 

[See fig. 54(b).] 

(a) (b) 

(a) Bevel gears. 

(b) Spur gears. 

Figure 58.-Simple differentials. 

Another type of differential (called the fixed type) has 

a large, fixed ratio. Such a drive is an evolution of the 

compound epicyclic train (fig. 54(b)). If the ring gear is 

replaced with a sun gear which meshes with a planet gear 

NP,R, the equations in table 22 apply except that the plus 

signs change to minus. If the sun and planet gears are 

made almost but not exactly equal, the output speed is 

the small difference between two terms that are almost 

equal. Such a drive is good for ratios from 10 to 3000. 

Using two ring gears instead of a ring and a sun in 

figure 54(b) results in a fixed-differential drive suitable 

for ratios of 15 to 100. Any epicyclic geartrain can be 

designed for differential operation. For instance, instead 

of one of the elements, such as a sun or ring gear, being 

fixed, two of the elements may be driven independently. 

Output speed is then the net result of the two inputs. 

Compound epicyclic trains can produce several input 

or output speeds by the addition of extra sun or ring gears 

meshing with the planet pinions. Thus the compound 

epicyclic train (fig. 54(b)) may have another sun gear 

Condition 

c Sun fixed 

Arm fixed 

Ring fixed 

Revolution ofSun I I Arm Ring 

meshing with the ring gear NP,R and another ring gear 

meshing with the sun gear Np,s. Any one of the four (two 

suns and two rings) could be fixed and the others 

available for input or output or free (ref. 14). 

9.6 Closed-Loop Trains 

Epicyclic geartrains with more than one planet pinion 

meshing with the same sun and ring gears have parallel 

paths through which power can flow. Other geartrains 

too sometimes use multiple-tooth contact to increase the 

capacity within a given space. Such gearing is sometimes 

called locked-train gearing. 

Two considerations arise with multiple-tooth contact 

that are not present in open trains. One is the proper 

selection of tooth numbers and spacing to ensure 

assembly. The other is gear accuracy and adjustment to 

ensure equal distribution of the load to each mesh. 

Multiple-mesh or locked-train gearing requires careful 

attention to tooth accuracy and support of the gears. 

Backlash and backlash tolerances should be as nearly 

equal as possible at each mesh to ensure equal load 

distribution (ref. 14). 

58 
Appendix-Symbols 

A 

a 

AB 

40 

b 

c 

AC 

Cl to c, 

Cl0 

c 

D 

DlO 

d 

E 

E’ 

e 

F 

Fe 

f 

G 

GlO 

H 

f!B 

AH 

h 

h 

I 

J 

K 

KC? 

Kb 

KL 

Knl 

KR 

KS 

KT 

KV 

KX 

k 

cone distance, m (in) 

semimajor axis of Hertzian contact ellipse, 

m (in) 

change in backlash, m (in) 

bearing life at 90-percent reliability level 

semiminor axis of Hertzian contact ellipse, 

m (in) 

distance between centers, m (in) 

change in center distance, m (in) 

special constants (table 19) 

special constant (eq. (68)) 

distance from neutral axis to outermost fiber 

of beam, m (in) 

diameter of pitch circle, m (in) 

special constant (eq. (79)) 

impingement depth, m (in) 

Young’s modulus of elasticity, Pa (psi) 

effective elastic modulus, Pa (psi) 

Weibull slope, dimensionless 

face width of tooth, m (in) 

effective face width, m (in) 

coefficient of friction 

dimensionless materials parameter 

life of gear at 90-percent reliability level, 

millions of revolutions or hr 

dimensionless film thickness 

Brine11 hardness 

differential hardness 

film thickness, m (in) 

central film thickness 

area moment of inertia, m4 (in4) 

geometry factor 

load intensity factor, Pa (psi) 

application factor 

rim thickness factor 

life factor 

load distribution factor 

reliability factor 

size factor 

temperature factor 

dynamic load factor 

cutter radius factor 

ellipticity ratio, dimensionless 

L 

LW 

e 

M 

mc 

“G 

N 

NC? 

n 

0 

P 

prl 

p* 

P 

Pa 

pb 

Pn 

Pt 

AP 

Q 

4 

Rc 

r 

ra 

r av 

‘b 

TC 

rP 

s 

s Q,t 

S w 

sb 

SC 

S, 

St 

8 

ST 

s 

T 

life, hr or millions of revolutions or stress 

cycles 

lead of worm gear, m (in) 

length, m (in) 

bending moment, N-m (lb-in) 

contact ratio 

gear ratio, Nz/Nr 

number of teeth 

equivalent number of teeth 

angular speed, rpm 

centerpoint location 

diametral pitch, m - 1 (in - 1) 

normal diametral pitch; m- 1 (in - 1) 

transverse diametral pitch, m - 1 (in - 1) 

circular pitch, m (in) 

axial pitch, m (in) 

base pitch, m (in) 

normal circular pitch, m (in) 

transverse circular pitch, m (in) 

oil jet pressure, Pa (psig) 

power loss 

number of planets 

Rockwell C scale hardness 

radius, m (in) 

addendum circle radius, m (in) 

mean pitch cone radius, m (in) 

base circle radius, m (in) 

cutter radius, m (in) 

pitch circle radius, m (in) 

stress, Pa (psi) 

allowable tensile stress, Pa (psi) 

allowable yield stress, Pa (psi) 

bending stress, Pa (psi) 

maximum Hertz compressive stress, Pa (psi) 

residual shear stress, Pa (psi) 

root fillet tensile stress, Pa (psi) 

probability of survival 

probability of survival of complete 

transmission including bearings and gears 

distance measured from pitch point along 

line of action, m (in) 

torque, N-m (lb-in) 

59 
TIO 

t 

u 

U 

V 

u 

vj 

ur - 

vr 

VS - 

US 

W 

WN 

wr 

tooth life at go-percent reliability, millions 

of stress cycles or hr 

tooth thickness, m (in) 

dimensionless speed parameter 

integer 

stressed volume, m3 (in3) 

pitch line velocity, m/s (ft/s) 

oil jet velocity, m/s (ft/s) 

total rolling velocity, m/s (ft/s) 

average total rolling velocity, m/s (ft/s) 

sliding velocity, m/s (ft/s) 

average sliding velocity, m/s (ft/s) 

dimensionless load parameter 

load normal to surface, N (lb) 

radial, or separating, component of load, 

N (lb) 

transmitted or tangential, component of 

load, N (lb) 

axial, or thrust, component of load, N (lb) 

semiwidth of contact path, m (in) 

distance from line of centers to impingement 

point, m (in) 

Lewis tooth form factor, dimensionless 

length of line of action, m (in) 

depth to critical shear stress, m (in) 

pressure-viscosity exponent, mVN (inUb) 

pitch angle, deg 

angle, deg 

tooth deflection, m (in) 

involute roll angle, deg (rad) 

pinion roll angle from base of involute to 

beginning of load (lowest point of double 

tooth-pair contact), deg (rad) 

increment of roll angle for which a single 

pair of teeth is in contact, deg (rad) 

increment of roll angle for which two pairs 

of teeth are in contact, deg (rad) 

life, millions of stress cycles 

angle of rotation, rad 

film thickness parameter, ratio of film 

thickness to rms surface roughness, 

dimensionless 

lead angle of worm gear, deg 

lubricant absolute viscosity, N s/m2 

(lb s/inz) 

absolute viscosity at standard atmospheric 

pressure, N s/m2 (lb s/in2) 

Poisson’s ratio, dimensionless 

efficiency 

radius of curvature, m (in) 

curvature difference, dimensionless 

shaft angle, deg 

curvature sum, m - 1 (in - 1) 

surface rms roughness, m (in) 

shear stress, Pa (psi) 

maximum shear stress, Pa (psi) 

maximum subsurface orthogonal reversing 

shear stress, Pa (psi) 

parameter defined in table 15 

pressure angle, deg 

normal pressure angle, deg 

transverse pressure angle, deg 

axial pressure angle, deg 

angle between load vector and tooth 

centerline (fig. 34), deg 

helix angle, deg 

spiral angle for spiral-bevel gear, deg 

angular velocity, rad/s 

Subscripts: 

a 

av 

B 

b 

eq 

G 

in 

j 

m 

max 

P 

P 

R 

r 

S 

s 

T 

t 

at addendum, at addendum circle 

average 

bearing 

binding 

equivalent 

gear 

input 

jet 

mesh 

maximum 

planet gear 

pinion 

ring gear 

rolling 

sun gear 

sliding 

total 

transverse 

60 
W 

W 

KY 

1 

2 

worm gear 

windage 

mutually perpendicular planes in which 

maximum and minimum (principal) radii of 

surface curvature lie 

pinion 

gear 

Superscripts: 

CA exponents 

CY pressure-viscosity exponent, mVN (inVlb) 

61 
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Analysis of Gear Tooth Bending Fatigue Strength. AGMA 

Technical Paper P229.24, American Gear Manufacturers 

Association, Oct. 1982. 

82. Chong, Tae-Hyong, et al.: Bending Stresses of Internal Spur 

Gear. Bull. JSME, vol. 25, no. 202, Apr. 1982, pp. 679-686. 

83. Wang, K.L.; and Cheng, H.S.: A Numerical Solution to the 

Dynamic Load, Film Thickness, and Surface Temperatures in 

Spur Gears, Part I-Analysis. J. Mech. Des., vol. 103, no. 1, Jan. 

1981, pp. 177-187. 

84. Wang, K.L.; and Cheng, H.S.: A Numerical Solution to the 

Dynamic Load, Film Thickness, and Surface Temperatures in 

Spur Gears, Part II-Results. J. Mech. Des., vol. 103, no. 1, Jan. 

1981, pp. 188-194. 

85. Dolan, T.J.; and Broghamer, E.L.: A Photoelastic Study of 

Stresses in Gear Tooth Fillets. University of Illinois, Engineering 

Experimental Station Bulletin 335, 1942. 

86. Cornell, R.W.: Compliance and Stress Sensitivity of Spur Gear 

Teeth. J. Mech. Des., vol. 103, no. 2, Apr. 1981, pp. 447-459. 

87. Coy, John J.; Townsend, Dennis P.; and Zaretsky, Erwin V.: 

Analysis of Dynamic Capacity of Low-Contact-Ratio Spur Gears 

Using Lundberg-Palmgren Theory. NASA TN D-8029, 1975. 

63 
88. Coy, John J.; and Zaretsky, Erwin V.: Life Analysis of Helical 

Gear Sets Using Lundberg-Palmgren Theory. NASA TN D-8045, 

1975. 

89. Coy, J.J.; Townsend, D.P.; and Zaretsky, E.V.: Dynamic 

Capacity and Surface Fatigue Life for Spur and Helical Gears. J. 

Lubr. Technol., vol. 98, no. 2, Apr. 1976, pp. 267-276. 

90. Townsend, D.P.; Coy, J.J.; and Zaretsky, E.V.: Experimental 

and Analytical Load-Life Relation for AISI 9310 Steel Spur Gears. 

J. Mech. Des., vol. 100, no. 1, Jan. 1978, pp. 54-60. 

91. Coy, John J.; Townsend, Dennis P.; and Zaretsky, Erwin V.: An 

Update on the Life Analysis of Spur Gears. Advanced Power 

Transmission Technology, George K. Fischer, ed., NASA 

CP-2210, 1983, pp. 421-433. 

92. Lundberg, G.; and Palmgren, A.: Dynamic Capacity of Rolling 

Bearings. Acta Polytech., Mech. Eng. Ser., vol. 1, no. 3, 1947. 

93. Bamberger, E.N., et al.: Life Adjustment Factors for Ball and 

Roller Bearings. American Society of Mechanical Engineers, 1971. 

94. Savage, M.; Paridon, CA.; and Coy, J.J.: Reliability Model for 

Planetary Gear Trains. Journal of Mechanisms, Transmissions, 

and Automation in Design, vol. 105, no. 3, Sept. 1983, pp. 

291-297. 

95. Krenzer, Theodore J.; and Knebel, Richard: Computer Aided 

Inspection of Bevel and Hypoid Gears, SAE Paper 831266, 1983. 

96. Litvin, F.L.; and Gutman, Y.: Methods of Synthesis and Analysis 

for Hypoid Gear-Drives of ‘Formate’ and ‘Helixform’, Parts 1, 2 

and 3. J. Mech. Des., vol. 103, no. 1, Jan. 81, pp. 83-113. 

97. Litvin, F.L.; and Gutman, Y.: A Method of Local Synthesis of 

Gears Grounded on the Connections Between the Principal and 

Geodetic Curvatures of Surfaces. J. Mech. Des., vol. 103, no. 1, 

Jan. 1981, pp. 114-125. 

98. Coy, J.J.; Rohn, D.A.; and Loewenthal, S.H.: Constrained 

Fatigue Life Optimization of a Nasvytis Multiroller Traction 

Drive. J. Mech. Des., vol. 103, no. 2, April 1981, pp. 423-429. 

99. Coleman, W.: Bevel and Hypoid Gear Surface Durability: Pitting 

and Scuffing. Source Book on Gear Design, Technology and 

Performance, Maurice A.H. Howes, ed., American Society for 

Metals, 1980, pp. 243-258. 

100. Anderson, W.J; and Zaretsky, E.V.: Rolling-Element Bearings. 

Mach. Des., vol. 42, no. 15, June 18, 1970, pp. 21-37. 

101. Ertel, A.M.: Hydrodynamic Lubrication Based on New 

Principles. Prikl. Mat. Mek., vol. 3, no. 2, 1939 (in Russian). 

102. Grubin, A.N.: Fundamentals of the Hydrodynamic Theory of 

Lubrication of Heavily Loaded Cylindrical Surfaces. Investigation 

of the Contact of Machine Components, Kh. F. Ketova, ed., 

Translation of Russian Book No. 30, Central Scientific Institute 

for Technology and Mechanical Engineering, Moscow, 1949. 

(Available from Dept. of Scientific and Industrial Research, Great 

Britain, Transl. CTS-235, and Special Libraries Assoc., Transl. 

R-3554.) 

103. Hamrock, Bernard J.; and Dowson, Duncan: Minimum Film 

Thickness in Elliptical Contacts for Different Regimes of FluidFilm Lubrication. NASA TP-1342, 1978. 

104. Fein, R.S.: Chemistry in Concentrated-Conjunction Lubrication. 

Interdisciplinary Approach to the Lubricaton of Concentrated 

Contacts, NASA SP-237, 1970, pp. 489-527. 

105. Godfrey, D.: Boundary Lubrication. Interdisciplinary Approach 

to Friction and Wear, NASA SP-181, 1968, pp. 335-384. 

106. Bowden, Frank Philip: and Tabor, David: The Friction and 

Lubrication of Solids, Vol. 2. Clarendon Press, Oxford, 1964, p. 

365. 

107. Borsoff, V.A.: Fundamentals of Gear Lubrication. Annual Rep., 

Shell Development Co., June 1955. (Work under Contract 

NO.a(s) 53-356-c.) 

108. Borsoff, V.A.; and Lulwack, R.: Fundamentals of Gear 

Lubrication. Final Rep., Shell Development Co., June 1957. 

(Work under Contract NO.a(s) 53-356-c). 

109. Horlick, E.J.; and O’D. Thomas, D. E.: Recent Experiences in the 

Lubrication of Naval Gearing. Gear Lubrication Symposium, 

Institute of Petroleum, 1966. 

110. Boner, Charles J.: Gear and Transmission Lubricant. Reinhold 

Pub. Co., 1964. 

111. Townsend, D.P.; and Akin, L.S.: Study of Lubricant Jet Flow 

Phenomena in Spur Gears-Out of Mesh Condition. Advanced 

Power Transmission Technology, George K. Fischer, ed., NASA 

CP-2210, 1983, pp. 461-476. 

112. Townsend, Dennis P.; and Akin, Lee S.: Gear Lubrication and 

Cooling Experiment and Analysis. Advanced Power Transmission 

Technology, George K. Fischer, ed., NASA CP-2210, 1983, pp. 

477-490. 

113. Akin, L.S.; Mross, J.J.; andTownsend, D.P.: Study of Lubricant 

Jet Flow Phenomena in Spur Gears. J. Lubr. Technol., vol. 97, 

no. 2, Apr. 1975, pp. 283-288, 295. 

114. Townsend, D.P.; and Akin, L.S.: Study of Lubricant Jet Flow 

Phenomena in Spur Gears-Out of Mesh Condition. J. Mech. 

Des., vol. 100, no. 1, Jan. 1978, pp. 61-68. 

115. Townsend, D.P.; and Akin, L.S.: “Analytical and Experimental 

Spur Gear Tooth Temperature as Affected by Operating 

Variables.” J. Mech. Des., vol. 103, no. 1, Jan. 1981, pp. 

219-226. 

116. Akin, L.S.; and Townsend, D.P.: Into Mesh Lubrication of Spur 

Gears with Arbitrary Offset Oil Jet. Part 1: For Jet Velocity Less 

Than or Equal to Gear Velocity. Journal of Mechanismis 

Transmissions, and Automation in Design, vol. 105, no. 4, Dec. 

1983, pp. 713-718. 

117. Akin, L.S.; and Townsend, D.P.: Into Mesh Lubrication of Spur 

Gears with Arbitrary Offset Oil Jet. Part 2: For Jet Velocities 

Equal to or Greater Than Gear Velocity. Journal of Mechanisms, 

Transmissions, and Automation in Design, vol. 105, no. 4, Dec. 

1983, pp. 719-724. 

118. Patir, Nadir; and Cheng, H.S.: Prediction of the Bulk 

Temperature in Spur Gears Based on Finite Element Temperature 

Analysis. ASLE Trans., vol. 22, no. 1, Jan. 1979, pp. 25-36. 

119. Anderson, Neil E.; and Loewenthal, S.H.: Spur-Gear-System 

Efficiency at Part and Full Load. NASA TP-1622, 1980. 

120. Anderson, N.E.; and Loewenthal, S.H.: Effect of Geometry and 

Operating Conditions on Spur Gear System Power Loss. J. Mech. 

Des., vol. 103, no. 1, Jan. 1981, pp. 151-159. 

121. Anderson, N.E.; and Loewenthal, S.H.: Design of Spur Gears for 

Improved Efficiency. J. Mech. Des., vol. 104, no. 3, Oct. 1982, 

pp. 767-774. 

122. Anderson, N.E.; and Loewenthal, S.H.: Selecting Spur-Gear 

Geometry for Greater Efficiency. Mach. Des., vol. 55, no. 10, 

May 12, 1983, pp. 101-105. 

123. Savage, M.; Coy, J.J.; and Townsend, D.P.: Optimal Tooth 

Numbers for Compact Standard Spur Gear Sets. J. Mech. Des., 

vol. 104, no. 4, Oct. 1982, pp. 749-758. 

124. Savage, Michael; Coy, John J.; and Townsend, Dennis P.: The 

Optimal Design of Standard Gearsets. Advanced Power 

Transmission Technology, George K. Fischer, ed., NASA 

CP-2210, 1983, pp. 435-469. 

125. Lynwander, P.: Gear Tooth Scoring Design Considerations. 

AGMA Paper P219.10, American Gear Manufacturers 

Association, Oct. 1981. 

126. Savage, M.; Coy, J.J.; and Townsend, D.P.: The Optimal Design 

of Involute Gear Teeth with Unequal Addenda. AVRADCOM TR 

82-C-7, NASA TM-82866, 1982. 

127. Carroll, R.K.; and Johnson, GE.: Optimal Design of Compact 

Spur Gear Sets. ASME Paper 83-DET-33, 1983. 

128. Wadlington, R.P.: Gear Drives. Mach. Des., vol. 39, no. 22, Sept. 

21, 1967, pp. 24-26. 

64 
129. DeBruyne, Neil A.: Design and Development Testing of Free 

Planet Transmission Concept. USAAMRDL-TR-74-27, Army 

Air Mobility Research and Development Laboratory, 1974. 

(AD-782857.) 

130. Folenta, D.J.: Design Study of Self-Aligning Bearingless 

Planetary Gear (SABP). Advanced Power Transmission 

Technology, George K. Fischer, ed., NASA CP-2210, 1983, pp. 

151-160. 

131. Nasvytis, Algirdas L.: Multiroller Planetary Friction Drives. SAE 

Paper 660763, Oct. 1966. 

132. White, G.: New Family of High-Ratio Reduction Gears with 

Multiple Drive Paths. Proc. Inst. Mech. Eng., vol. 188, no. 23, 

1974, pp. 281-288. 

133. Lastine, J.L.; and White, G.: Advanced Technology VTOL Drive 

Train Configuration Study. USAAVLABS-TR-69-69, Army 

Aviation Material Laboratories, 1970. (AD-867905.) 

134. White, G.: Helicopter Transmission Arrangements with SplitTorque Gear Trans. Advanced Power Transmission Technology, 

George K. Fischer, ed., NASA CP-2210, 1983, pp. 141-150. 

65 
Bibliography 

Buckingham, E.: Analytical Mechanics of Gears. Dover Publications, 

NY, 1963. 

Dudley, D. W.: Gear Handbook. McGraw-Hill, NY, 1962. 

Boner, C. J.: Gear and Transmission Lubricants. Reinhold Publishing, 

NY, 1964. 

Khiralla, T. W.: On the Geometry of External Involute Spur Gears. 

T. W. Khiralla, Studio City, Calif., 1976. 

Deutschman, A.; Michels, W. J.; and Wilson, C. E.: Machine Design, 

MacMillan, NY, 1975, pp. 519-659. 

Michalec, G. W.: Precision Gearing: Theory and Practice. John Wiley 

& Sons, NY, 1966. 

Dudley, D. W.: Practical Gear Design. McGraw-Hill, NY, 1954. 

Mechanical Drives Reference Issue, Machine Design, 3rd ed., Vol. 39, 

No. 22, Sept. 21, 1967. 

66 
1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No. 

NASA RP-1152 AVSCOM TR 84-C-15 

4. Title and Subtitle 5. Report Date 

Gearing December 1985 

6. Performing Organization Code 

7. Author(s) 6. Performing Organization Report No. 

John 3. Coy, Dennis P. Townsend, and E-2003 

Erwin V. Zaretsky 

10. Work Unit No. 

9. Performing Organization Name and Address 

Propulsion Directorate, USAARTA-AVSCOM 11. Contract or Grant No. 

and 

Lewis Research Center 

Cleveland, Ohio 44135 13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Reference Publication 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 14. Sponsoring Agency Code 

and 

U.S. Army Aviation Systems Command 

St. Louis, MO 63120-1798 

15. Supplementary Notes 

John J. Coy: Propulsion Directorate, U.S. Army Aviation Research and Technology 

Activity (USAARTA-AVSCOM), Lewis Research Center, Cleveland, Ohio. 

Dennis P. Townsend and Erwin V. Zaretsky: Lewis Research Center, Cleveland, Ohio. 

Ch. 38 in Mechanical Design and Systems Handbook, McGraw-Hill Book Company, 1985. 

16. Abstract 

Gearing technology in its modern form has a history of only 100 years. However, 

the earliest form of gearing can probably be traced back to fourth century B.C. 

Greece. This publication draws together current gear practice and integrates It 

with recent advances in the technology. The history of gearing 'IS reviewed 

brlefly in the Introduction. Subsequent sections describe types of gearing and 

their geometry, processing, and manufacture. Both conventional and more recent 

methods of determining gear stress and deflections are considered. The subjects 

of life prediction and lubrication are additions to the literature, New and more 

complete methods of power loss prediction as well as an optimum design of spur 

gear meshes are described. Conventional and new types of power transmission 

systems are presented. 

7. Key Words (Suggested by Author(s)] 16. Distribution Statement 

Gearing; Gear geometry; Gear processing; Unclassified - unlimited 

Gear stress; Gear deflection STAR Category 37 

9. Security Classif. (of this report) 20. Security Classif. (of this page) 

Unclassified 

21. No. of pages 22. Price 

Unclassified 72 A04 

For sale by the National Technical Information Service, Springfield, Virginia 22161 

NASA-Langley, 1985